Jürg
Kramer
HU Berlin
Humboldt Arithmetic Geometry Seminar 📅
Institute
Head
Jürg Kramer
Usual time
Tuesdays from 13:15 to 14:45
Usual venue
Room 3.006, Rudower Chaussee 25, 12489 Berlin
Number of talks
325
July 2024
Remy
van Dobben de Bruyn
Utrecht
June 2024
Gari Yamel
Peralta Alvarez
HU Berlin
Toric arithmetic varieties and adelic intersection numbers
Abstract.
A result of Burgos, Philippon and Sombra states that the height of a toric arithmetic variety with respect to a line bundle equipped with a torus-invariant continuous semipositive metric is given by the integral of a convex function over a polytope. In this talk, I will discuss a generalization of this formula for line bundles with torus-invariant singular metrics. The techniques used to obtain this result are based on Yuan and Zhang's theory of adelic line bundles. Moreover, this formula agrees with the generalized adelic intersection numbers of Burgos and Kramer.
Marco
Flores
HU Berlin
Modularity of formal Fourier-Jacobi series from a cohomological point of view
Abstract.
In the Kudla program, generating series constructed by means of arithmetic intersection numbers are conjectured to appear as Fourier coefficients of modular forms. One instance of this program was the proof in 2015 by Bruinier and Raum of Kudla's modularity conjecture over \(\mathbb{C}\) for special cycles on a Shimura variety of orthogonal type, obtained by showing that certain formal series of Jacobi forms are in fact modular forms. This result was reformulated and proven over \(\mathbb{Z}\) by Kramer using the Faltings-Chai theory of toroidal compactifications of \(\mathcal{A}_g\). In this talk, we contextualize the above result using cohomology of line bundles on such a toroidal compactification \(\overline{\mathcal{A}}_g\), which we believe is the more natural point of view. We then obtain cohomological vanishing results characterizing the modularity of formal Fourier-Jacobi series, taking advantage of the natural resolution morphism from \(\overline{\mathcal{A}}_g\) to the minimal compactification \(\mathcal{A}^*_g\).
May 2024
Paul
Ziegler
Darmstadt
p-adic integration on Artin stacks
Abstract.
After giving an introduction to the technique of p-adic integration, I will explain how this technique can be extended to Artin stacks, and give an application to BPS invariants. This is joint work with M. Groechenig and D. Wyss.
Emiliano
Ambrosi
Strasbourg
Reduction modulo p of the Noether problem
Abstract.
Let k be an algebraically closed field of characteristic \( p\ge 0 \) and V a faithful k-rational representation of an \(l\)-group G. Noether's problem asks whether V/G is (stably) birational to a point. If \( l = p \), then Kuniyoshi proved that this is true, while for \( l\neq p \) Saltman constructed \(l\)-groups for which V/G is not stably rational. Hence, the geometry of V/G depends heavily on the characteristic of the field. We show that for all the groups G constructed by Saltman, one cannot interpolate between the Noether problem in characteristic 0 and p. More precisely, we show that there does not exist a complete valuation ring R of mixed characteristic (0,p) and a smooth proper R-scheme \( X \to \mathrm{Spec}(R) \) whose special fiber and generic fiber are both stably birational to V/G. The proof combines the integral p-adic Hodge theoretic results of Bhatt-Morrow-Scholze, with the study of the Cartier operator on differential forms in positive characteristic. This is a joint work with Domenico Valloni.
Victor
Gonzalez Alonso
Hannover
Infinitesimal rigidity of certain modular morphisms
Abstract.
The Torelli morphism maps (the isomorphism class of) a smooth complex projective curve to its polarized jacobian variety. It has been recently proved by Farb that this is the only non-constant holomorphic map from the moduli space of curves to that of principally polarized abelian varieties, and Serván has recently proved a similar result for the Prym morphism. These result can be interpreted by saying that a certain moduli space of morphisms consists of just one point, and it is natural to ask whether this point is reduced. In this talk I will present a joint work with Giulio Codogni and Sara Torelli, where we show that this is indeed the case (in the setting of moduli stacks): These morphisms do not admit non-trivial infinitesimal deformations. The proof uses the Fujita decomposition of the Hodge bundle of a family of curves, and can be applied to other morphisms involving moduli of smooth curves.
April 2024
Alessandro
Ghigi
Pavia
Projective structures on Riemann surfaces and metrics on the moduli space of curves
Abstract.
I will describe some recent results on projective structures on Riemann surfaces. After recalling some basic definitions I will explain a correspondence between varying projective structures over the moduli space of curves and (1,1)-forms over it. I will describe explicitely the correspondence in two examples: the projective structure coming from uniformization and a projective structure coming from Hodge theory. Finally I will also describe a new projective structure obtain from the line bundle \( 2 \Theta\).
February 2024
Walter
Gubler
Regensburg
Abstract divisorial spaces and an extension of adelic
intersection numbers
Abstract.
This is joint work in progress with Yulin Cai. Yuan and Zhang defined an adelic intersection theory over number fields and Yuan used this to give a striking new approach to the uniform Mordell-Lang approach. Recently, Burgos and Kramer extended the arithmetic intersection pairing allowing more singular metrics on the archimedean side. We complete the picture on the non-archimedean side. Using the framework of so called abstract divisorial spaces, we show that Yuan-Zhang's construction is a completion process which works in various situations. In particular, we can extend arithmetic intersection numbers allowing more singular metrics working over any reasonable base field with product formula. In particular, we can do that for proper adelic base curves in the framework of Chen and Moriwaki.
January 2024
Annette
Werner
Frankfurt
The Hodge-Tate sequence for commutative rigid analytic groups
Abstract.
We consider generalizations of Scholze's Hodge-Tate sequence on smooth, proper rigid analytic varieties. These generalizations feature coefficients in commutative rigid groups, which are locally p-divisible. We will also discuss applications to p-adic versions of Simpson's correspondence with coefficients in commutative rigid groups. This is joint work with Ben Heuer and Mingjia Zhang.
Fabien
Pazuki
Copenhagen
Explicit bounds on the coefficients of modular polynomials and
the size of \(X_0(N)\)
Abstract.
We give explicit upper and lower bounds on the size of the coefficients of the modular polynomials for the elliptic \(j\)-function. These bounds make explicit the best previously known asymptotic bounds. The proof relies on a careful study of the Mahler measure of a family of specializations of the modular polynomial. We also give an asymptotic comparison between the Faltings height of the modular curve \(X_0(N)\) and the height of this modular polynomial, giving a link between these two ways of measuring the "size" of the modular curve. The talk is based on joint work with Florian Breuer and Desirée Gijon Gomez.
Michele
Pernice
KTH Stockholm
A stacky Castelnuovo’s contraction theorem
Abstract.
In this talk, we are going to discuss a generalization to weighted blow-ups of the classical Castelnuovo's contraction theorem. Moreover, we will show as a corollary that the moduli stack of n-pointed stable curves of genus 1 is a weighted blow-up. This is a joint work with Arena, Di Lorenzo, Inchiostro, Mathur, Obinna.
December 2023
Bruno
Kahn
IMJ Paris
Universal Weil cohomology
Abstract.
In this joint work with Luca Barbieri-Viale, we show that a universal Weil cohomology exists over any field k. The story is actually a bit more complicated: to a suitable class of smooth projective k-varieties (all varieties is the default) we associate 4 universal Weil cohomologies, depending on whether the universal problem concerns targets which are additive or abelian categories, and whether the axioms for the Weil cohomology are plain or if one adds requirements in the style of Weak and Strong Lefschetz. In the latter case, the universal additive category obtained can be used to recover André’s category of motives for ''motivated'' cycles. If time permits, I will explain how the construction extends over a base, and some open problems.
Gergely
Berczi
Aarhus
Geometry of the Hilbert scheme of points on manifolds, part I
Abstract.
While the Hilbert scheme of points on surfaces is well-explored and extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be exceptionally intricate, exhibiting wild and pathological characteristics. Our understanding of their topology and geometry is very limited. In this series of two talks I will present recent results on various aspects of their geometry. I will discuss i) the distribution of torus fixed points within the components of the Hilbert scheme (work with Jonas Svendsen), ii) intersection theory of Hilbert schemes: a new formula for tautological integrals over geometric components, and applications in enumerative geometry, iii) new link invariants of plane curve (and hypersurface) singularities coming from p-adic integration and Igusa zeta functions (work with Ilaria Rossinelli). Part II of the talk will be relatively independent from part I and takes place on Wednesday 13 December in the Algebraic Geometry Seminar.
Gabriel
Dill
Bonn
The modular support problem over number fields and over function
fields
Abstract.
In 1988, Erdős asked: let \(a\) and \(b\) be positive integers such that for all \(n\), the set of primes dividing \(a^n - 1\) is equal to the set of primes dividing \( b^n - 1\). Is \(a = b\)? Corrales and Schoof answered this question in the affirmative and showed more generally that, if every prime dividing \(a^n - 1\) also divides \(b^n - 1\), then \(b\) is a power of \(a\). In joint work with Francesco Campagna, we have studied this so-called support problem with the Hilbert class polynomials \( H_D(T)\) instead of the polynomials \(T^n - 1\), replacing roots of unity by singular moduli. I will present the results we obtained both in the number field case, where \(a\) and \(b\) lie in some ring of \(S\)-integers in a number field \(K\), as well as in the function field case, where \(a\) and \(b\) are regular functions on a smooth irreducible affine curve over an algebraic closure of a finite field.
November 2023
Tobias
Kreutz
Bonn
On Simpson's Standard Conjecture for unipotent local systems
Abstract.
Simpson's Standard Conjecture predicts that a local system which is defined over \( \overline{\mathbb{Q}}\) on both sides of the Riemann-Hilbert correspondence is motivic. In this talk, I want to discuss this conjecture for unipotent local systems. Conditional on classical transcendence conjectures for mixed Tate motives over number fields, we show that a unipotent local system over \( X = \mathbb{P}^1 \setminus \{s_1,...,s_n\}\) which is defined over \(\overline{\mathbb{Q}}\) on both sides of the Riemann-Hilbert correspondence is the monodromy of a mixed Tate motive over \(X\). This uses the construction of the motivic fundamental group due to Deligne-Goncharov, Borel's computation of the (rational) algebraic \(K\)-theory of number fields, and a homotopy exact sequence for the motivic fundamental group due to Esnault-Levine. For unipotent local systems of small index, we obtain some unconditional results due to transcendence/irrationality results of Baker and Apéry. We also prove a version of the Standard Conjecture over smooth projective curves of genus one, using transcendence results of Chudnovsky and Wüstholz.
Luigi
Lombardi
Milano
On the invariance of Hodge numbers of irregular varieties under derived equivalence
Abstract.
A conjecture of Orlov predicts the invariance of the Hodge numbers of a smooth projective complex variety under derived equivalence. For instance this has been verified to the case of varieties of general type. In this talk, I will examine the case of varieties that are fibered by varieties of general type through the Albanese map. For this class of varieties I will prove the derived invariance of Hodge numbers of type \( h^{0,p}\), together with a few other invariants arising from the Albanese map. This talk is based on a joint work with F. Caucci and G. Pareschi.
October 2023
Erwan
Rousseau
Brest
An Albanese construction for Campana's C-pairs
Abstract.
We will explain a construction of Albanese maps for orbifolds (or C-pairs), with applications to hyperbolicity such as a generalization of the Bloch-Ochiai theorem. (Joint with Stefan Kebekus).
July 2023
Jan Hendrik
Bruinier
TU Darmstadt
Special cycles on toroidal compactifications of orthogonal Shimura
varieties
Abstract.
A famous theorem of Gross-Kohnen-Zagier states that the generating series of Heegner divisors on a modular curve is a weight 3/2 modular form with values in the first Chow group. An analogous result for special divisors on orthogonal Shimura varieties was proved by Borcherds, and for higher codimension special cycles by Zhang, Raum and myself. We report on joint work with Shaul Zemel generalizing the modularity result to special divisors on toroidal compactifications of orthogonal Shimura varieties.
Hsueh-Yung
Lin
National Taiwan University
Motivic invariants of birational maps and Cremona groups
Abstract.
(Joint with E. Shinder) In characteristic zero, birational maps of projective varieties factorize through a sequence of blow-ups and blow-downs along smooth centers. We study to which extent these factorization centers are unique, and construct invariants of motivic nature which account for the non-uniqueness of centers. For surfaces over a perfect field, we prove (with E. Shinder and S. Zimmermann) the uniqueness of centers in the strongest possible sense. In higher dimension, we construct examples showing that the centers fail to be unique. Relying on the non-uniqueness, we provide new explanations of the non-simplicity of Cremona groups.
June 2023
Per
Salberger
U Gothenburg
On n-torsion in class groups of number fields
Abstract.
It is well known that the class group of a number field is of size bounded above by roughly the square root of its discriminant. But one expects by conjectures of Cohen-Lenstra that the the n-torsion part of this group should be much smaller and there have recently been several papers on this by prominent mathematicians. We present in our talk some of these results and a partial improvement of an estimate of Bhargava, Shankar, Taniguchi, Thorne, Zimmerman and Zhao.
Michael
McBreen
U Hong Kong
Introduction to Microlocal Sheaves
Abstract.
This is the first of a series of lectures about microlocal sheaves. Details can be found here.
Haohao
Liu
IMJ Paris
Fourier-Mukai transform on complex tori
Abstract.
Classical Fourier transform occupies a major part of the analysis. An analog on abelian varieties is introduced by S. Mukai in 1981, which is now known as Fourier-Mukai transform. Similar to the Fourier inversion formula, Mukai proved a duality theorem for his transform. This result reveals the phenomenon that, the derived category of coherent modules of two non-isomorphic projective varieties can be equivalent. In this talk, I will present the work of O. Ben-Bassat, J. Block and T. Pantev about the analytic version of Fourier-Mukai transform on complex tori.
May 2023
Josh
Lam
HU Berlin
Properties of non-abelian Hodge theory mod p: Periodicity
Abstract.
This is part of the reading group on flat connections in positive and mixed characteristic. For details see here.
Alexandre
Minets
U Edinburgh
A proof of the P=W conjecture
Abstract.
Let \( C \) be a smooth projective curve. The non-abelian Hodge theory of Simpson is a homeomorphism between the character variety \( M_B \) of \( C \) and the moduli of (semi)stable Higgs bundles \( M_D \) on \( C \). Since this homeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weight filtration. Based on computations in small rank, de Cataldo-Hausel-Migliorini conjectured that the weight filtration on \( H^*(M_B) \) gets sent to the perverse filtration on \( H^*(M_D) \), associated to the Hitchin map. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on \( H^*(M_D) \). Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.
Emil
Jacobsen
U Zürich / Stockholm University
On (generic) motivic fundamental groups
Abstract.
I will introduce the (generic) motivic fundamental group of a smooth variety, and explain its relation to the usual (generic) fundamental group. The main result can also be phrased as follows: generic local systems of motivic origin are stable under extension in the category of generic local systems. I will also present a (generic) motivic version of a classical theorem of Hain, on Malcev completions of monodromy representations. At the end, I'll explain some of the group/representation theoretic tools that go into these result. If there's time, I can explain some analogous Hodge theoretic results, as well as some future directions.
Marco
Flores
HU Berlin
A cohomological approach to formal Fourier-Jacobi series
Abstract.
Siegel modular forms admit various expansions, one of the most important being the Fourier-Jacobi expansion. Algebraically, these expansions take the form of a series whose coefficients are Jacobi forms satisfying a certain symmetry condition. One poses the following modularity question: does every formal series of this shape arise from a Siegel modular form? Bruinier and Raum answered the question affirmatively, over the complex numbers, in 2014. In this talk I will consider this question over the ring of integers, and reformulate it as a matter of cohomological vanishing on a toroidal compactification of the moduli space A_g of principally polarized abelian varieties. I will present a proof that said vanishing is equivalent to our modularity question, and explore its relationship with the singularities of the minimal compactification of A_g.
April 2023
Ruijie
Yang
MPI Bonn
The Riemann-Schoktty problem and Hodge theory
Abstract.
It is a classical problem, going back to Riemann, to decide which principally polarized abelian varieties come from the Jacobians of curves. In 2008, Casalaina-Martin proposed a question towards the Riemann-Schottky problem in terms of the codimension of multiplicity locus of the theta divisor, which includes Debarre’s conjecture as a special case. In this talk, I will talk about a partial affirmative answer to this question using a newly developed theory “Higher multiplier ideals”, which builds on Sabbah-Schnell’s theory of complex Hodge modules and Beilinson-Bernstein’s language of twisted D-modules. It is based on joint work with Christian Schnell.
February 2023
Gari Yamel
Peralta Alvarez
HU Berlin
An adelic formula of Chern-Weil type for the height of a toric variety
Abstract.
The philosophy behind height functions is to measure the complexity of objects. These functions have proven to be extremely useful when dealing with finitude statements of arithmetic objects. Gillet and Soulé developed an arithmetic version of intersection theory which produces height functions on arithmetic varieties. These heights depend on the choice of a smooth hermitian line bundle. While theoretically satisfactory, writing down smooth metrics can be extremely difficult. In the context of Shimura varieties, the natural metrics to consider are singular. This calls for an extension of the theory that admits these examples. Recently, Yuan and Zhang gave a general framework that is compatible with known extensions. In this talk, we work out the case of toric varieties. This case can be described explicitly, and we can compute heights for singular metrics with a formula that resembles Chern-Weil theory.
January 2023
Martín
Sombra
ICREA and Universitat de Barcelona
The zero set of the independence polynomial of a graph
Abstract.
In statistical mechanics, the independence polynomial of a graph \(G\) arises as the partition function of the hard-core lattice gas model on \(G\). The distribution of the zeros of these polynomials when \( G \to +\infty\) is relevant for the study of this model and, in particular, for the determination of its phase transitions. In this talk, I will review the known results on the location of these zeros, with emphasis on the case of rooted regular trees of fixed degree and varying depth \(k \ge 0\). In an on-going work with Juan Rivera-Letelier (Rochester, USA) we show that for these graphs, the zero sets of their independence polynomials converge as \(k \to \infty\) to the bifurcation measure of a certain family of dynamical systems on the Riemann sphere. In turn, this allows to show that the pressure function of this model has a unique phase transition, and that it is of infinite order.
December 2022
Moritz
Kerz
Regensburg
The Picard-Lefschetz formula for normal crossings
Abstract.
In the study of semi-stable degeneration of Lefschetz pencils one is led to a generalization of the classical Picard-Lefschetz formula for certain perverse sheaves on normal crossing spaces. In the talk I will recall the formalism of nearby cycle and vanishing cycle functors and I will explain how Hodge theory allows one to obtain the normal crossing Picard-Lefschetz formula. Joint work with A. Beilinson and H. Esnault.
Roberto
Gualdi
Regensburg
How to guess the height of the solutions of a system of
polynomial equations
Abstract.
A beautiful result due to Bernstein and Kushnirenko allows to predict the number of solutions of a system of Laurent polynomial equations from the combinatorial properties of the defining Laurent polynomials. In a joint work with Martín Sombra (ICREA and Universitat de Barcelona), we give intuitions for an arithmetic version of such a theorem. In particular, in the easy case of the planar curve x + y + 1 = 0, we show how to guess the height of its intersection with a twist of itself by a torsion point. The talk will involve the Arakelov geometry of toric varieties, special values of the Riemann zeta function and, unexpectedly, the most famous detective of the world literature.
November 2022
Laurentiu
Maxim
Wisconsin
On the topology of aspherical complex projective manifolds and related questions
Abstract.
I will report on recent progress on various open problems involving aspherical complex projective manifolds, including the Singer-Hopf conjecture and the Bobadilla-Kollar conjecture. The first part will be very informal, stating the main problems, historical developments, and some recent results. In the second part, I will introduce the main technical tools and sketch proofs of some of the recent results. Note: This is the first of a series of two talks. The second talk will be in the algebraic geometry seminar on Wednesday, see here.
Siddarth
Mathur
Orsay
Searching for the impossible Azumaya algebra
Abstract.
In two 1968 seminars, Grothendieck used the framework of étale cohomology to extend the definition of the Brauer group to all schemes. Over a field, the objects admit a well-known algebro-geometric description: They are represented by \( \mathbb{P}^n\)-bundles (equivalently: Azumaya Algebras). Despite the utility and success of Grothendieck's definition, an important foundational aspect remains open: Is every cohomological Brauer class over a scheme represented by a \( \mathbb{P}^n\)-bundle? It is not even known if smooth proper threefolds over the complex numbers have enough Azumaya algebras! In this talk, I will outline a strategy to construct a Brauer class that cannot be represented by an Azumaya algebra. Although the candidate is algebraic, the method will leave the category of schemes and use formal-analytic line bundles to create Brauer classes. I will then explain a strange criterion for the existence of a corresponding Azumaya Algebra. At the end, I will reveal the unexpected conclusion of the experiment.
Annette
Huber-Klawitter
Freiburg
Period Numbers
Abstract.
Period numbers are complex numbers like \( \pi, \log(2) \) or \( \zeta(5)\). They can be described as values of integrals. As apparent from the examples, such numbers appear in many places in mathematics and are of great interest in transcendence theory. More recently, a variant involving also the exponential function has come into focus. I am going to explain the definition and its conceptual interpretation.
Riccardo
Pengo
MPI Bonn
On the Northcott property for special values of L-functions
Abstract.
According to Northcott's theorem, each set of algebraic numbers whose height and degree are bounded is finite. Analogous finiteness properties are also satisfied by many other heights, as for instance the Faltings height. Given the many (expected and proven) links between heights and special values of L-functions (with the BSD conjecture as the most remarkable example), it is natural to ask whether the special values of an L-functions satisfy a Northcott property. In this talk, based on a joint work with Fabien Pazuki, and on another joint work in progress with Jerson Caro and Fabien Pazuki, we will show how this Northcott property is often satisfied at the left of the critical strip, and not satisfied on the right. We will also overview the links between these Northcott properties and those of the motivic heights defined by Kato, and also some effective aspects of our work, which aim at giving some explicit bounds for the cardinality of the finite sets that we come across.
Constantin
Podelski
HU Berlin
The degree of the Gauss map for certain Prym varieties
and the Schottky problem
Abstract.
One approach to the Schottky problem is to study the stratification of the moduli space \( \mathcal{A}_g \) of principally polarized abelian varieties by the dimension of the singular locus of the theta divisor: Andreotti and Mayer have shown that the locus of Jacobians is an irreducible component of the stratum \( \mathcal{N}_{g-4} \subset \mathcal{A}_g\). In a similar spirit one can stratify \(\mathcal{A}_g \) by the degree of the Gauss map: Recently, Codogni and Krämer have shown that the locus of Jacobians is also an irreducible component of the corresponding Gauss stratum. Motivated by this, we study the Gauss map on various other irreducible components of \(\mathcal{N}_{g-4}\) that parametrize certain Prym varieties, and we show that for each \( g \ge 4\) one of these components has the same Gauss degree as Jacobians.
October 2022
Marco
D'Addezio
Jussieu
Hecke orbits on Shimura varieties
Abstract.
I will talk about the proof of the Hecke orbit conjecture, proposed by Chai and Oort. I will mainly focus on two of the tools that are used. The first main ingredient is a new local result on the monodromy groups of F-isocrystals, which enhances Crew's parabolicity conjecture. Another one is the Cartier-Witt stack constructed by Bhatt-Lurie. This is a joint work with Pol van Hoften.
July 2022
Ya
Deng
U Lorraine
Gari Yamel
Peralta Alvarez
HU Berlin
Arithmetic intersection theory on toric varieties with singular metrics
Abstract.
In the context of arithmetic intersection theory, the height of a variety with respect to a hermitian line bundle is an arithmetic analogue of the usual degree. Gillet and Soulé have developed a general construction for line bundles equipped with smooth metrics. In many interesting examples, the natural metrics to consider are not smooth. This called for extensions of this theory that admit such metrics. In this talk we focus on the case of toric varieties, for which Burgos, Philippon and Sombra gave an explicit description that admits continuous semipositive metrics. We sketch a generalization of these results to a broader type of singularities. This is based on the theory of adelic line bundles of Yuan and Zhang.
June 2022
Fabrizio
Barroero
U Roma Tre
On the Zilber-Pink conjecture for complex abelian varieties and distinguished categories
Abstract.
The Zilber-Pink conjecture is a very general statement that implies many well-known results in diophantine geometry, e.g., Manin-Mumford, Mordell-Lang and André-Oort. I will report on recent joint work with Gabriel Dill in which we proved that the Zilber-Pink conjecture for a complex abelian variety A can be deduced from the same statement for its trace, i.e., the largest abelian subvariety of A that can be defined over the algebraic numbers. This gives some unconditional results, e.g., the conjecture for curves in complex abelian varieties (over the algebraic numbers this is due to Habegger and Pila) and the conjecture for arbitrary subvarieties of powers of elliptic curves that have transcendental j-invariant. While working on this project we realised that many definitions, statements and proofs were formal in nature and we came up with a categorical setting that contains most known examples and in which (weakly) special subvarieties can be defined and a Zilber-Pink statement can be formulated. We obtained some conditional as well as some unconditional results.
José Ignacio
Burgos Gil
ICMAT
Chern-Weil theory and Hilbert-Samuel theorem for semi-positive
singular toroidal metrics on line bundles
Abstract.
In this talk I will report on joint work with A. Botero, D. Holmes and R. de Jong. Using the theory of b-divisors and non-pluripolar products we show that Chern-Weil theory and a Hilbert Samuel theorem can be extended to a wide class of singular semi-positive metrics. We apply the techniques relating semipositive metrics on line bundles to b-divisors to study the line bundle of Siegel-Jacobi forms with the Peterson metric. On the one hand we prove that the ring of Siegel-Jacobi forms of constant positive relative index is never finitely generated, and we recover a formula of Tai giving the asymptotic growth of the dimension of the spaces of Siegel-Jacobi modular forms.
Christopher
Deninger
U Münster
Dynamical systems for arithmetic schemes
Abstract.
For any arithmetic scheme \(X\) we construct a continuous time dynamical system whose periodic orbits come in compact packets that are in bijection with the closed points of \( X \). All periodic orbits in a given packet have the same length equal to the logarithm of the order of the residue field of the corresponding closed point. For \( X = \mathrm{Spec}\, \mathbb{Z}\) we get a dynamical system whose periodic orbits are related to the prime numbers. The construction uses new ringed spaces which are constructed from rational Witt vector rings. In the zero-dimensional case we recover a construction of Kucharczyk and Scholze who realized certain Galois groups as étale fundamental groups of ordinary topological spaces. A p-adic version of our construction turns out to be closely related to the Fargues-Fontaine curve of p-adic Hodge theory.
Gabriel
Ribeiro
École Polytechnique Paris
Cartier duality, character sheaves, and generic vanishing
Abstract.
Since Green and Lazarsfeld's seminal work, generic vanishing theorems have influenced many fields ranging from birational geometry to analytic number theory. In this talk, I will present a new generic vanishing theorem for holonomic D-modules, that may shed new light on both of these worlds. Following ideas of Laumon, I'll explain how "character sheaves" play the role of topologically trivial line bundles and construct their moduli space based on a stacky version of Cartier duality.
Morten
Lüders
U Hannover
Milnor K-theory of p-adic rings and motivic cohomology
Abstract.
We explain a joint work with Matthew Morrow on \(p\)-adic Milnor K-theory. Our main theorem is a comparison of mod \(p^r\) Milnor K-groups of \(p\)-henselian local rings with the Milnor range of a newly defined syntomic cohomology theory by Bhatt, Morrow and Scholze. We begin by putting our result into context. Then we sketch the proof which builds on an analysis of a filtration on Milnor K-groups and a new technique called the left Kan extension from smooth algebras.
May 2022
Sarah
Zerbes
ETH Zürich
Euler systems and the Birch—Swinnerton-Dyer conjecture for abelian surfaces
Abstract.
Euler systems are one of the most powerful tools for proving cases of the Bloch-Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture. I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for \( \mathrm{GSp}(4)\), and an explicit reciprocity law relating the Euler system to values of \( L\)-functions. I will then explain recent work with Loeffler, where we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over \( \mathbb{Q} \), and for modular elliptic curves over imaginary quadratic fields.
Marco
Flores
HU Berlin
A cohomological approach to formal Fourier-Jacobi series
Abstract.
Siegel modular forms admit various expansions, one of the most important being the Fourier-Jacobi expansion. Algebraically, these expansions take the form of a series whose coefficients are Jacobi forms satisfying a certain symmetry condition. One then poses the following modularity question: does every formal series of that shape come from a Siegel modular form? Bruinier and Raum answered the question affirmatively, over the complex numbers, in 2014. In this talk I will consider this question over the ring of integers, and reformulate it as a matter of cohomological vanishing. I will present a weaker version of the desired cohomological vanishing, and a result highlighting how special the case of genus g=2 potentially is.
Dougal
Davis
U Edinburgh
Mixed Hodge modules on flag varieties and representations of real reductive groups
Abstract.
In this talk, I will give a gentle introduction to applications of mixed Hodge modules in the representation theory of real reductive groups. Since the work of Beilinson-Bernstein, Kazhdan-Lusztig and Lusztig-Vogan, it has been understood how to realise representations of real groups in terms of twisted D-modules on flag varieties, and how weight filtrations coming from the corresponding mixed sheaves over a finite field can be used to gain strong control over their structure. Much more recently, it was suggested by Schmid and Vilonen that passing instead to mixed Hodge modules reveals more information about representations than is accessible over a finite field, through Hodge filtrations and polarisations. I will sketch this broad story, including the main conjecture of Schmid-Vilonen relating Hodge structures to unitarity, and, as time permits, explain some concrete results appearing in recent joint work of myself and Vilonen (arXiv:2202.08797) on the computation of the Hodge structures in Kazhdan-Lusztig-Vogan theory and their connection with the unitarity algorithm of Adams-van Leeuwen-Trapa-Vogan.
Marco
Maculan
IMJ Paris
Counting rational points on varieties with large fundamental group
Abstract.
A nonsingular projective curve of genus at least 2 on a number field admits only finitely many rational points. Elliptic curves might have infinitely many rational points (as the projective line does), but “way less” than the projective line. In a joint work with Y. Brunebarbe, inspired by a recent result of Ellenberg-Lawrence-Venkatesh, we prove an analogous statement in higher dimension: projective varieties with large fundamental group in the sense of Kollár-Campana have “way less” rational points than Fano varieties.
February 2022
Alex
Torzewski
King's College London
Lawrence--Venkatesh bounds for curves in families
Abstract.
We outline how the method of Lawrence-Venkatesh to bound rational points via p-adic period mappings can be used in families. This leads to upper bounds on the number of rational points on curves of genus > 1 depending only on the reduction modulo a well chosen prime and the primes of bad reduction. This was first shown by Faltings as a consequence of the Mordell and Shafarevich Conjectures.
Klaus
Künnemann
Regensburg
The Calabi-Yau problem in archimedean and non-archimedean geometry
Abstract.
We discuss the Calabi-Yau problem on complex manifolds and its analog in non-archimedean geometry. The complex Calabi-Yau problem asks for solutions of a PDE of Monge-Ampère type. It was posed by Calabi in 1954 and solved by Yau in 1978. After a short reminder of the complex case we give a brief introduction to non-archimedean analytic geometry and report on recent progress in the non-archimedean case.
January 2022
Tobias
Kreutz
HU Berlin
Arithmetic and $\ extit{l}$-adic aspects of special subvarieties
Abstract.
We define an $\ell$-adic analog of the Hodge theoretic notion of a special subvariety. The Mumford-Tate conjecture predicts that the two notions are equivalent. In this talk, I want to discuss some properties of these subvarieties and prove this equivalence for subvarieties satisfying a certain monodromy condition. This builds on work of Klingler, Otwinowska and Urbanik on the fields of definition of special subvarieties.
Christian
Lehn
TU Chemnitz
Singular varieties with trivial canonical class
Abstract.
We will present recent advances in the field of singular varieties with trivial canonical class obtained in joint work with Bakker and Guenancia building on work by many others. This includes the decomposition theorem which says that such a variety is up to a finite cover isomorphic to a product of a torus, irreducible Calabi-Yau (ICY) and irreducible symplectic varieties (ISV). The proof uses a reduction argument to the projective case which in turn is possible due to advances in deformation theory and a certain result about limits of Kähler Einstein metrics in locally trivial families.
Benedikt
Preis
Regensburg
Motivic nearby cycle functors and local monodromy
Abstract.
We give a generalisation of Grothendieck's local monodromy theorem. The proof is completely independent of Grothendieck's classical proofs and shows that the unipotence phenomenon is of motivic nature.
Mads
Villadsen
Stony Brook
Generic vanishing and Chen-Jiang decompositions
Abstract.
The Generic Vanishing theorem of Green and Lazarsfeld describes the behaviour of the cohomology of direct images of canonical bundles to abelian varieties when twisted by line bundles with vanishing first Chern class. I will discuss Chen-Jiang decompositions of these direct images, first introduced by J. Chen and Z. Jiang for generically finite morphisms to abelian varieties, which explain their generic vanishing behaviour and certain positivity properties in detail. In particular, I will discuss how to prove the existence of these decompositions using classical variational Hodge theory.
December 2021
Constantin
Podelski
HU Berlin
The degree of the Gauss map on Theta divisors
Abstract.
The Gauss map attaches to any smooth point of a theta divisor in an abelian variety its tangent space translated to the origin. For indecomposable principally polarized varieties this is a generically finite map. In this talk, I will compute its degree for a generic ppav on some irreducible components of Andreotti-Mayer loci that have been introduced by Debarre in terms of specific Prym constructions. The computation will rely on the technique of Lagrangian specialisation.
November 2021
Igor
Burban
U Paderborn
Tame non-commutative nodal curves, gentle algebras and homological mirror symmetry
Abstract.
In my talk, I am going to introduce a class of finite dimensional associative algebras, which are derived equivalent to certain tame non-commutative nodal curves. Following the approach of Lekili and Polishchuk, the constructed derived equivalence allows to interpret such non-commutative curves as homological mirrors of appropriate graded compact oriented surfaces with non-empty marked boundary. This is a joint work with Yuriy Drozd.
Antareep
Mandal
HU Berlin
Uniform sup-norm bounds for Siegel cusp forms
Abstract.
Let $\Gamma\subsetneq \mathrm{Sp}(n,\mathbb{R})$ be a torsion-free arithmetic subgroup of the symplectic group $\mathrm{Sp}(n,\mathbb{R})$ acting on the Siegel upper half-space $\mathbb{H}_n$ of degree $n$. Consider the $d$-dimensional space of Siegel cusp forms $\mathcal{S}_k^n(\Gamma)$ of weight $k$ for $\Gamma$ and let $\{f_j\}_{1\leq j\leq d}$ be a basis of $\mathcal{S}_k^n(\Gamma)$ orthonormal with respect to the Petersson inner product. In this talk we present our work regarding the sup-norm bound of the quantity $S_k^{\Gamma}(Z):=\sum_{j=1}^{d}\det (Y)^{k}\vert{f_j(Z)}\vert^2\,(Z\in\mathbb{H}_n)$. We show using the heat kernel method, for $n=2$ unconditionally and for $n>2$ subject to a conjectural determinant-inequality, that $S_k^{\Gamma}(Z)$ is bounded above by $c_{n,\Gamma} k^{n(n+1)/2}$ when $M:=\Gamma\backslash\mathbb{H}_n$ is compact and by $c_{n,\Gamma} k^{3n(n+1)/4}$ when $M$ is non-compact of finite volume, where $c_{n,\Gamma}$ denotes a positive real constant depending only on the degree $n$ and the group $\Gamma$. Furthermore, we show that this bound is uniform in the sense that if we fix a group $\Gamma_0$ and take $\Gamma$ to be a subgroup of $\Gamma_0$ of finite index, then the constant $c_{n,\Gamma}$ in these bounds depends only on the degree $n$ and the fixed group $\Gamma_0$.
Ruijie
Yang
HU Berlin
Vanishing cycles for divisors
Abstract.
Given a holomorphic function $f$ on $\mathbb{C}^n$ with an isolated critical point at the origin, the vanishing cycles associated to $f$ are homology cycles of the level set $\{f=t\}$ for $|t|\ll 1$. If the singularity is non-isolated, the vanishing cycles glue to complexes of vector spaces. In algebraic geometry, divisors arise more frequently than functions. A natural question is, how do the vanishing cycles associated to local defining functions glue together? In this talk I will explain a global construction using $\mathscr{D}$-modules and discuss its relation to singularities of divisors. This is based on joint work in progress with Christian Schnell.
Salvatore
Floccari
U Hannover
Isogenous hyper-Kähler varieties
Abstract.
The Torelli theorem for hyper-Kähler varieties explains to which extent such a variety can be recovered from its integral second cohomology, together with its pairing and Hodge structure. I will address a variant of this problem: How much of a hyper-Kähler variety is determined by its rational second cohomology? Work of Huybrechts and Fu-Vial provides the answer for K3 surfaces. In higher dimension we expect this rational cohomology group to control the full motive of the variety; the main result of the talk confirms this in the realm of André motives.
February 2020
Adrian
Langer
U Warsaw
On smooth projective $\mathscr{D}$-affine varieties
Abstract.
A $\mathscr{D}$-affine variety is a variety on which left $\mathscr{D}$-modules behave as on an affine variety. Unlike in the case of $\mathscr{O}$-modules there exist smooth projective varieties that are $\mathscr{D}$-affine (e.g., rational homogeneous varieties in characteristic zero). I will survey known results on smooth projective $\mathscr{D}$-affine varieties. In particular, I will classify $\mathscr{D}$-affine smooth projective surfaces. In positive characteristic, a basic tool that I use is a new generalization of Miyaoka's generic semipositivity theorem.
January 2020
Ezra
Waxman
TU Dresden
Random Models for the Distribution of Primes with a Prescribed Primitive Root
Abstract.
The Hardy-Littlewood Conjecture (also known as the k-tuple conjecture) is a vast generalization of the twin prime conjecture. In particular, it gives an asymptotic count for the number of integer pairs (n,n+d) such that both n and n+d are prime. In this talk, I will discuss preliminary results concerning the application of these heuristic models to the distribution of primes with a prescribed primitive root. Specifically, for integers \\(g\\) \geq 2\\) and \\(d \\in 2\mathbb{N}\\), I will present a conjecture for the number of prime pairs \\(p,p+d\\) such that \\(g\\) is a primitive root modulo both \\(p\\) and \\(p+d\\). Time permitting, I will also introduce preliminary findings concerning the distribution of primes with a prescribed primitive root, across short intervals. Joint work with Magdaléna Tinková & Mikuláš Zindulka.
Antareep
Mandal
HU Berlin
Sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
This talk updates our progress towards obtaining a sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. We begin by generalizing the relation between the cusp forms and the Maass forms in higher dimensions. Then we obtain the spherical functions for the Laplace-Beltrami operator in Siegel upper half-space by reduction to the complex case and make a weight correction to make them correspond to the Siegel-Maass Laplacian. Then we use these to construct the heat kernel, whose analysis leads to the weight dependence of the sup-norm bound.
Dragos
Fratila
U Strasbourg
The Jordan-Chevalley decomposition for semistable $G$-bundles on elliptic curves
Abstract.
For projective curves of arithmetic genus one, there are three possibilities: nodal curve, cuspidal curve, elliptic curve. For $G$ a reductive group one can consider semistable $G$-bundles on such a curve and their moduli stack. In the nodal case one recovers the adjoint quotient $G/G$ and in the cuspidal case the Lie algebra version of the adjoint quotient. The elliptic curve case could be considered as a further "exponentiation" of the group $G$. We will review the Jordan-Chevalley decomposition in the Lie algebra and Lie group case and then proceed to explain how one can formulate such a decomposition for $G$-bundles over an elliptic curve. Then we'll see that the Jordan-Chevalley decomposition can be also expressed in terms of the existence of a certain stratification of the moduli stack of semistable $G$-bundles. In the nodal or cuspidal case the strata are described in terms of nilpotent cones of certain Levi or pseudo-Levi subgroups. I will explain a similar description in the elliptic case. Finally, I hope to explain how such a stratification can be used to study certain automorphic sheaves on $\mathrm{Bun}_G(E)$ for an elliptic curve in analogy to the study of character sheaves for groups or Lie algebras. This is joint work with Sam Gunningham and Penghui Li.
December 2019
Anil
Aryasomayajula
IISER Tirupati, India
On the holomorphic version of a conjecture by Sarnak
Abstract.
In 1995, Iwaniec and Sarnak computed estimates of Hecke Eigen Maass forms associated to co-compact arithmetic subgroups of $\mathrm{SL}(2,\mathbb{R})$, and Sarnak went on to make a conjecture on the growth of Hecke Eigen Maass forms. Adapting the arguments of Iwaniec and Sarnak to the setting of Hecke Eigen cusp forms, we discuss a holomorphic version of the conjecture of Sarnak.
Martin
Gallauer
U Oxford
Simplicity of Tannakian categories, and applications
Abstract.
In this talk, I want to discuss two classification problems in algebraic geometry: (1) Given a variety, classify its constructible sheaves (in the derived sense). (2) Given a field, classify its motives (in the derived sense). One aspect which connects the two problems is the appearance of Tannakian categories (in the derived sense). I will draw attention to the fact that these are "simple", and explain how this allows us to make progress on the two classification problems.
November 2019
Thorsten
Heidersdorf
U Bonn
Tensor categories in positive characteristic
Abstract.
Some of the most important tensor categories over a field come from representations of algebraic groups. A celebrated theorem of Deligne asserts that every tensor category of subexponential growth is the representation category of an algebraic supergroup scheme. The theorem is no longer true in characteristic $p > 0$. Counterexamples arise from representations of the cyclic group $\mathbb{Z}/p\mathbb{Z}$. Ostrik proposed a conjectural extension, but recently Benson-Etingof constructed an infinite chain of counterexamples in characteristic 2. These categories are closely related to representations of algebraic groups and the question if monoidal categories admit abelian envelopes. I will give an overview about these results and discuss some recent developments.
Marco
Flores
HU Berlin
Cohomological dimension of projective schemes in pro-p towers
Abstract.
If $X$ is a variety of dimension $d$ over an algebraically closed field, its étale cohomology groups with coefficients in any constructible sheaf vanish in degree greater than $2d$. Moreover, if $X$ is affine they already vanish in degree greater than $d$ by Artin's vanishing theorem. The last is not true for projective varieties. However, Scholze showed that if $X$ is a complex projective variety of dimension $d$ and $p$ is a prime number, there is a specific tower of $p$-power degree covers of $X$ such that the direct limit of étale cohomology groups with $ extbackslash mathbb{F}_p$-coefficients does vanish in degree greater than $d$. In this talk we present a new proof of this result, by Hélène Esnault, which moreover works over any algebraically closed field of characteristic different from $p$.
Ziyang
Gao
IMJ-PRG Paris
Functional transcendence on the universal abelian variety
Abstract.
The main topic of this talk is the mixed Ax-Schanuel theorem for the universal abelian variety. I will explain the statement and sketch its proof. We will start from the analogous statement for abelian varieties over the field of complex numbers.
October 2019
Yuri
Bilu
U Bordeaux
Diversity in Parametric Families of Number Fields
Abstract.
Let $X$ be a projective curve over $\mathbb{Q}$ of genus $g$ and $t$ a non-constant $\mathbb{Q}$-rational function of degree $m>1$. For every $n\in \mathbb{N}$, pick $P_n\in X$ with $t(P_n)=n$. Hilbert's Irreducibility Theorem (HIT) says that for infinitely many $n$ the field $\mathbb{Q}(P_n)$ is of degree $m$ over $\mathbb{Q}$. Moreover, this holds for overwhelmingly many $n$: Among the number fields $\mathbb{Q}(P_1), \dots ,\mathbb{Q}(P_n)$ there is only $o(n)$ fields of degree less than $m$. However, HIT does not say how many distinct field there are among $\mathbb{Q}(P_1), ... ,\mathbb{Q}(P_n)$. A 1994 result of Dvornicich and Zannier implies that for large $n$, there are at least $cn/\log n$ distinct among those fields, with $c=c(m,g)>0$. Conjecturally there should be a positive proportion (that is, $cn$) of distinct fields. This conjecture is proved in many special cases in the work of Zannier and his collaborators, but in general, getting rid of the log term seems very hard. We make a little step towards proving this conjecture. While we cannot remove the log term altogether, we can replace it by log n raised to a power strictly smaller than 1. To be precise, we prove that for large $n$ there are at least $n/(\log n)^{1-e}$ distinct fields, where $e=e(m,g)>0$. A joint work with Florian Luca.
July 2019
Grégoire
Menet
U Grenoble
Integral cohomology of quotients via toric geometry
Abstract.
I will provide some new methods, based on toric blow-ups, to determine the integral cohomology of complex manifolds quotiented by automorphisms groups of prime order. Indeed, quotient singularities can locally be interpreted as toric varieties, and the framework of toric geometry is well adapted to deal with integral cohomology. The original motivation to study this problem was the computation of an important invariant in the context of hyperhähler geometry: the Beauville-Bogomolov form.
Yuri
Bilu
U Bordeaux
Singular units do not exist
Abstract.
It is classically known that a singular modulus (a j-invariant of a CM elliptic curve) is an algebraic integer. Habegger (2015) proved that at most finitely many singular moduli are units, answering a question of Masser (2011). However, his argument, being non-effective, did not imply any bound for the size of these "singular units". I will report on a recent work with Philipp Habegger and Lars Kühne, where we prove that singular units do not exist at all. First, we bound the discriminant of any singular unit by 10^15. Next, we rule out the remaining singular units using computer-assisted arguments.
June 2019
Ana Maria
Botero
U Regensburg
The convex set algebra and the $b$-Chow ring of toric varieties
Abstract.
We extend McMullen's polytope algebra to the so called convex set algebra. We show that the convex set algebra embeds in the projective limit of the Chow cohomology rings of all smooth toric compactifications of a given torus, with image generated by the classes of all nef toric $b$-divisors. It follows that the convex set algebra can be viewed as a universal ring for intersection theory of nef toric $b$-cocycles on the toric Riemann-Zariski space. We further discuss some applications of this viewpoint towards a combinatorial interpretation of non-archimidean Arakelov theory of toric varieties over discretely valued fields in the sense of Bloch-Gillet-Soulé.
Giulio
Orecchia
U Rennes
a tropical/monodromy criterion for existence of Néron models
Abstract.
Néron models are central objects to the study of degenerations of abelian varieties over Dedekind schemes. However, over bases of higher dimension, they do not always exist. In this talk I will introduce a criterion, called toric additivity, for an abelian family with semistable reduction to admit a Néron model. It can be expressed in terms of monodromy action on the $ extbackslash$ell-adic Tate module, and, in the case of jacobians of curves, in terms of finiteness of the tropical jacobian.
Antonio
Rojas-Leon
U Sevilla
Local Systems with sporadic monodromy groups
Abstract.
Let $X$ be a curve over a separably closed field $k$ of characteristic $p$ and $F$ an $ extbackslashell$-adic local system on $X$, where $\ell \neq p$. If we view this local system as a representation of $\pi_1(X)$, its Zariski closure is the global monodromy group $G$ of $F$. When $X$ and $F$ are defined over a finite field, this group determines (after a normalization) the distribution of the Frobenius traces of $F$ on the points of $X$ with values over sufficiently large finite extensions of the base field. In general, one expects this group to be as large as allowed for by the geometric properties of $F$. In particular, only in exceptional cases it will be finite. Abhkanyar's conjecture determines which finite groups can appear as monodromy groups of such local systems. In particular, if $G$ is simple finite, it can be the monodromy group of a local system on the affine line in characteristic $p$ if and only if $p$ divides the order of $G$. In this talk we will give some naturally constructed examples of local systems on the affine line and the punctured affine line on small characteristic whose monodromy groups are sporadic finite: the Conway groups Co1, Co2, Co3, the Suzuki group 6.Suz or the McLaughlin group McL. This is joint work with Nicholas M. Katz (Princeton) and Pham H. Tiep (Rutgers).
Patrick
Graf
U Bayreuth
Reflexive differential forms in positive characteristic
Abstract.
Given a differential form on the smooth locus of a normal variety defined over a field of positive characteristic, we discuss under what conditions it extends to a resolution of singularities (possibly with logarithmic poles). Our main result works for log canonical surface pairs over a perfect field of characteristic at least seven. We also give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. If time permits, we will give applications to the study of the Lipman-Zariski conjecture.
Quentin
Guignard
ENS Paris
Geometric $\ell$-adic local factors
Abstract.
I will explain how to give a cohomological definition of epsilon factors for $\ell$-adic sheaves over a henselian trait of positive equicharacteristic distinct from $\ell$. The resulting formula is reminiscent of the cohomological construction by Katz of the $\ell$-adic Swan representation, and involves Gabber-Katz extensions as well. These local factors provide a product formula for the determinant of the cohomology of an $\ell$-adic sheaf on a curve over a field of positive characteristic distinct from $\ell$. When the base field is finite, this specializes to the classical theory of Dwork, Langlands, Deligne, and Laumon.
May 2019
Ezra
Waxman
U Prague
Hecke Characters and the $L$-Function Ratios Conjecture
Abstract.
A Gaussian prime is a prime element in the ring of Gaussian integers $\mathbb{Z}[i]$. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Hecke proved that the Gaussian primes are equidistributed across sectors of the complex plane by making use of Hecke characters and their associated $L$-functions. In this talk I will present several applications obtained upon applying the $L$-functions Ratios Conjecture to this family of $L$-functions. In particular, I will present a refined conjecture for the variance of Gaussian primes across sectors, and a conjecture for the one level density across this family.
Johan
Commelin
U Freiburg
On the cohomology of smooth project surfaces with $p_g = q = 2$ and maximal Albanese dimension
Abstract.
In this talk I will report on a joint project with Matteo Penegini (Genova). The second cohomology of a surface S as mentioned in the title splits up as a sum of two pieces. One piece comes from the Albanese variety. The other piece looks like the cohomology of a K3 surface, which we call a K3 partner X of S. If the surface S is a product-quotient then we can geometrically construct the K3 partner X and an algebraic correspondence that relates the cohomology of S and X. Finally, we prove the Tate and Mumford-Tate conjectures for all surfaces S that lie in the same connected component of the Gieseker moduli space as a product-quotient surface.
April 2019
Giulio
Bresciani
FU Berlin
Essential dimension and pro-finite group schemes
Abstract.
The essential dimension of an algebraic group is a measure of the complexity of its functor of torsors, i.e. $H^1(--,G)$. Classically, essential dimension has only been studied for group schemes of finite type. We study the case of pro-finite group schemes, and prove two very general criteria that show that essential dimension is almost always infinite for pro-finite group schemes: thus, it does not provide much information about them. We thus propose a new, natural refinement of essential dimension, the fce dimension. The fce dimension coincides with essential dimension for group schemes of finite type but has a better behaviour otherwise. Over any field, we compute the fce dimension of the Tate module of a torus. Over fields finitely generated over Q, we compute the fce dimension of Z_p and of the Tate module of an abelian variety.
February 2019
Hsueh-Yung
Lin
U Bonn
Algebraic approximations of compact Kähler manifolds of algebraic codimension 1
Abstract.
Let $X$ be a compact Kähler manifold. The so-called Kodaira problem asks whether $X$ has arbitrarily small deformations to some projective varieties. In this talk, we will explain our solution to the Kodaira problem for compact Kähler manifolds of algebraic dimension $a(X) = dim(X) − 1$. This is partly joint work with B. Claudon and A. Höring.
January 2019
Jan Hendrik
Bruinier
TU Darmstadt
Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series
Abstract.
Let $V$ be a rational quadratic space of signature $(m,2)$. A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with $SO(V)$ to the coefficients of the central derivative of a Siegel Eisenstein series of genus $m+1$. We report on joint work with Tonghai Yang proving this conjecture for the coefficients of non-singular index $T$ under certain conditions on $T$. To this end we establish some new local arithmetic Siegel-Weil formulas at the archimedean and non-archimedean places.
Paola
Frediani
U Pavia
On the geometry of some special subvarieties contained in the Torelli locus
Abstract.
Almost all the examples of special subvarieties of $A_g$ contained in the Torelli locus are given by families of Jacobians of Galois covers of the projective line or of elliptic curves. They satisfy a sufficient condition that I will explain. I will first show that this condition is also necessary in the case of double covers of elliptic curves. In fact I will prove that the bielliptic locus is not totally geodesic for $g>3$. Finally I will discuss the geometry of some examples of Galois covers of elliptic curves yielding special subvarieties of $A_3$, which are fibered in totally geodesic curves and hence contain countably many Shimura curves.
Renjie
Lyu
U Amsterdam
A result of algebraic cycles on cubic hypersurfaces
Abstract.
The Chow group of algebraic cycles of a smooth projective variety is an important subject in algebraic geometry, which in general, is too massive to grasp. In this talk, we show that the Chow group of a smooth cubic hypersurface $X$ can be recovered by the algebraic cycles of its Fano variety of lines $F(X)$. It generalizes $M$. Shen’s previous work of 1-cycles on cubics. The proof relies on some birational geometry concerning the Hilbert square of cubic hypersurfaces recently studied by E. Shinder, S. Galkin and C. Voisin. As applications, when $X$ is a complex smooth 4-fourfold, the result we obtained could prove the integral Hodge conjecture for 1-cycles on the polarised hyper-Kähler variety $F(X)$. In the arithmetic aspect, C. Schoen addressed the integral analog of the Tate conjecture, which is predicted to be true for 1-cycles of any smooth projective variety defined over finite fields. We will show how to use our result to prove this conjecture for 1-cycles on the Fano variety $F(X)$ if $X$ is a smooth cubic 4-fold over a finitely generated field.
December 2018
Christian
Liedtke
TU München
Rigid rational curves in positive characteristic
Abstract.
Rational curves are central to higher-dimensional algebraic geometry. If a rational curve “moves” on a variety, then the variety is uniruled and in characteristic zero, this implies that the variety has negative Kodaira dimension. Over fields of positive characteristic, varieties can be inseparably uniruled without having negative Kodaira dimension. However, I will show in my talk that in the case that a rational curve moves on a surface of non-negative Kodaira dimension, then this rational curve must be “very singular”. In higher dimensions, there is a similar result that is more complicated to state. I will also give examples that show the results are optimal. This is joint work with Kazuhiro Ito and Tetsushi Ito.
Marco
D'Addezio
FU Berlin
Finiteness of perfect torsion points of an abelian variety and $F$-isocrystals
Abstract.
I will report on a joint work with Emiliano Ambrosi. Let $k$ be a field which is finitely generated over the algebraic closure of a finite field. As a consequence of the theorem of Lang-Néron, for every abelian variety over $k$ which does not admit any isotrivial abelian subvariety, the group of $k$-rational torsion points is finite. We show that the same is true for the group of torsion points defined on a perfect closure of $k$. This gives a positive answer to a question posed by Hélène Esnault in 2011. To prove the theorem we translate the problem into a certain question on morphisms of $F$-isocrystals. Then we handle it by studying the monodromy groups of the $F$-isocrystals involved.
Jürg
Kramer
HU Berlin
Sup-norm bounds of automorphic forms
Abstract.
In our talk we will talk about new approaches for establishing optimal sup-norm bounds for Maass forms.
November 2018
Philipp
Reichenbach
TU Berlin
Vector bundles on elliptic curves and an associated Tannakian category
Abstract.
In 1957 Atiyah classified all vector bundles on elliptic curves for an algebraically closed ground field. Moreover, for the characteristic 0 case he completely described the multiplicative structure, i.e. the behavior of the tensor product. In this talk we review the essential results due to Atiyah and will interpret them in the light of Tannakian categories. Namely, allowing only morphisms of vector bundles on elliptic curves that respect the Harder-Narasimhan filtration leads to a neutral Tannakian category. For characteristic 0 we discuss some properties of the corresponding affine group scheme and give complete classifications of certain Tannakian sub-categories. Finally, some known results for the characteristic $p$ case are stated and questions for future research will be formulated.
Thorsten
Herrig
HU Berlin
Fixed points and entropy of endomorphisms on complex tori
Abstract.
We investigate the fixed-point numbers and entropies of endomorphisms on complex tori. Motivated by an asymptotic perspective that has turned out in recent years to be so fruitful in Algebraic Geometry, we study how the number of fixed points behaves when the endomorphism is iterated. In this talk I show that the fixed-points function can have only three kinds of behaviour, and I characterize them in terms of the analytic eigenvalues. An interesting follow-up question is to determine criteria to decide of which type an endomorphism is. I will provide such criteria for simple abelian varieties in terms of the possible types of endomorphism algebras. The gained insight into the occurring eigenvalues can be applied to questions about the entropy of an endomorphism. I will give criteria for an endomorphism to be of zero or positive entropy and answer the important question whether the entropy can be the logarithm of a Salem number.
Peter
Jossen
ETH Zürich
Exponential Periods and Exponential Motives
Abstract.
I begin by explaining Nori's formalism and how to use it to construct abelian categories of motives. Then, following ideas of Katz and Kontsevich, I show how to construct a tannakian category of "exponential motives" by applying Nori's formalism to rapid decay cohomology, which one thinks of as the Betti realisation. This category of exponential motives contains the classical mixed motives à la Nori. We then introduce the de Rham realisation, as well as a comparison isomorphism with the Betti realisation. When k = IQ, this comparison isomorphism yields a class of complex numbers, "exponential periods", which includes special values of the gamma and the Bessel functions, the Euler-Mascheroni constant, and other interesting numbers which are not expected to be periods of classical motives. In particular, we attach to exponential motives a Galois group which conjecturally governs all algebraic relations among their periods.
October 2018
Nero
Budur
KU Leuven
Absolute sets and the Decomposition Theorem
Abstract.
The celebrated Monodromy Theorem states that the eigenvalues of the monodromy of a polynomial are roots of unity. In this talk we give an overview of recent results on local systems giving a generalization of the Monodromy Theorem. We end up with a conjecture of André-Oort type for special loci of local systems. If true in general, it would provide a simple conceptual proof for all semi-simple perverse sheaves of the Decomposition Theorem, assuming only the geometric case of perverse sheaves constructed from the constant sheaf (Beilinson-Bernstein-Deligne-Gabber). We prove the conjecture in rank one. Thus we have a new proof of the Decomposition Theorem for perverse sheaves constructed from rank one local systems. Joint work with Botong Wang.
Giulio
Codogni
U Roma Tre
Gauss map, singularities of the theta divisor and trisecants
Abstract.
The Gauss map is a finite rational dominant map naturally defined on the theta divisor of an irreducible principally polarised abelian varieties. In the first part of this talk, we study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. Thanks to this analysis, we obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus. We spell out this bound in several examples, and we use it to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension. This is a joint work with S. Grushevsky and E. Sernesi. In the second part of this talk, we will study the relation between the Gauss map and trisecant of the Kummer variety. Fay's trisecant formula shows that the Kummer variety of the Jacobian of a smooth projective curve has a four dimensional family of trisecant lines. We study when these lines intersect the theta divisor of the Jacobian, and prove that the Gauss map of the theta divisor is constant on these points of intersection, when defined. We investigate the relation between the Gauss map and multisecant planes of the Kummer variety as well. This is a joint work with R. Auffarth and R. Salvati Manni.
July 2018
Javier
Fresán
École Polytechnique
Hodge Theory of Kloosterman Connections
Abstract.
Broadhurst and Roberts recently studied the L-functions associated with symmetric powers of Kloosterman sums and conjectured a functional equation after extensive numerical computations. By the work of Yun, these L-functions correspond to “usual” motives over Q which, in low degree, are known to be modular. For the purpose of computing the Hodge numbers or relating the L-functions to periods, it is however more convenient to change gears and work with exponential motives. I will construct the relevant motives and show how the irregular Hodge filtration allows one to explain the gamma factors at infinity in the functional equation, as well as to get lower bounds for the p-adic valuations of Frobenius eigenvalues. It is a joint work with Claude Sabbah and Jeng-Daw Yu.
June 2018
Alberto Navarro
Garmendia
Uni Zürich
Recent advancements on the Grothendieck-Riemann-Roch theorem
Abstract.
Grothendieck's Riemann-Roch theorem compares direct images at the level of K-theory and the Chow ring. After its initial proof at the Borel-Serre report, Grothendieck aimed to generalise the Riemann-Roch theorem at SGA 6 in three directions: allowing general schemes not necessarily over a base field, replacing the smoothness condition on the schemes by a regularity condition on the morphism, and avoiding any projective assumption either on the morphism or on the schemes. After the coming of higher K-theory there was also a fourth direction, to prove the Riemann-Roch also between higher K-theory and higher Chow groups. In this talk we will review recent advancements in these four directions during the last years. If time permits, we will also discuss refinements of the Riemann-Roch formula which takes into account torsion elements.
Giuseppe
Ancona
Uni Strasbourg
On the standard conjecture of Hodge type for abelian fourfolds
Abstract.
Let S be a surface and V be the Q-vector space of divisors on S modulo numerical equivalence. The intersection product defines a non degenerate quadratic form on V. We know since the Thirties that it is of signature (1,d-1), where d is the dimension of V. In the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is an easy consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable. Moreover, using a classical product formula on quadratic forms, the p-adic result will give us non-trivial informations on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.
Anna
von Pippich
TU Darmstadt
An analytic class number type formula for PSL2(Z)
Abstract.
For any Fuchsian subgroup Γ ⊂ PSL2(R) of the first kind, Selberg introduced the Selberg zeta function in analogy to the Riemann zeta function using the lengths of simple closed geodesics on Γ\H instead of prime numbers. In this talk, we report on a formula that determines the special value at s = 1 of the derivative of the Selberg zeta function for Γ = PSL2(Z). This formula is obtained as an application of a generalized Riemann--Roch isometry for the trivial sheaf on Γ\H, equipped with the Poincaré metric. This is joint work with Gerard Freixas.
May 2018
Barbara
Jung
HU Berlin
The arithmetic self intersection number of the Hodge bundle on A2
Abstract.
The Hodge line bundle ω on the moduli stack A2, metrized by the L2-metric, can be identified with the bundle of Siegel modular forms, metrized by the logarithmically singular Petersson metric. Intersection theory for line bundles on arithmetic varietes, developed by Gillet and Soulé and generalized to the case of line bundles with logarithmically singular metrics by Burgos, Kramer, and Kühn, states a formula for the arithmetic self intersection number ϖ4 in terms of integrals of Green currents over certain cycles on the complex fibre of A2, and a contribution from the finite fibres. The computation of these ingredients for ϖ4 can be tackled by regarding the space A2 from different points of view. We will specify these viewpoints and sketch the associated methods for obtaining the explicit value of ϖ4.
Roberto
Laface
TU München
Picard numbers of abelian varieties in all characteristics
Abstract.
I will talk about the Picard numbers of abelian varieties over C (joint work with Klaus Hulek) and over fields of positive characteristic. After providing an algorithm for computing the Picard number, we show that the set Rg of Picard numbers of g-dimensional abelian varieties is not complete if g ≥ 2, that is there exist gaps in the sequence of Picard numbers seen as a sequence of integers. We will also study which Picard numbers can or cannot occur, and we will deduce structure results for abelian varieties with large Picard number. In characteristic zero we are able to give a complete and satisfactory description of the overall picture, while in positive characteristic there are several pathologies and open questions yet to be answered.
Robin
de Jong
Uni Leiden
The Néron-Tate heights of cycles on abelian varieties
Abstract.
Given a polarized abelian variety A over a number field and an effective cycle Z on A, there is naturally attached to Z its so-called Néron-Tate height. The Néron-Tate height is always non-negative, and behaves well with respect to multiplication-by-N on A. Its good properties have led and still lead to applications in number theory. We discuss formulas for the Néron-Tate heights of some explicit cycles. First we focus on the tautological cycles on jacobians, where we find a new proof of the Bogomolov conjecture for curves. Then we focus on the symmetric theta divisors on a general principally polarized abelian variety. Here we find an explicit relation with the Faltings height. The latter part is based on joint work with Farbod Shokrieh.
Emre
Sertöz
MPI Leipzig
Computing and using periods of hypersurfaces
Abstract.
The periods of a complex projective manifold X are complex numbers, typically expressed as integrals, which give an explicit representation of the Hodge structure on the cohomology of X. Although they provide great insight, periods are often very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will give a different algorithm which can compute the periods of any smooth projective hypersurface and can do so with much higher precision. As an application, we will demonstrate how to reliably guess the Picard rank of quartic K3 surfaces and the Hodge rank of cubic fourfolds from their periods.
April 2018
Ulf
Kühn
Uni Hamburg
Multiple q-zeta values and period polynomials
Abstract.
We present a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we will state a dimension conjecture for the spaces of their weight- and depth-graded parts, which have a similar shape as the conjectures of Zagier and Broadhurst-Kreimer for multiple zeta values.
February 2018
Yuri
Bilu
U Bordeaux
Equations with singular moduli: effective aspects
Abstract.
A singular modulus is the j-invariant of an elliptic curve with complex multiplication. André (1998) proved that a polynomial equation F(x,y)=0 can have only finitely many solutions in singular moduli (x,y), unless the polynomial F(x,y) is 'special' in a certain precisely defined sense. Pila (2011) extended this to equations in many variables, proving the André-Oort conjecture on C^n. The arguments of André and Pila were non-effective (used Siegel-Brauer). I will report on a recent work by Allombert, Faye, Kühne, Luca, Masser, Pizarro, Riffaut, Zannier and myself about partial effectivization of these results.
Xavier
Roulleau
U Aix-Marseille
Irrationality of cubic threefolds by their reduction mod 3
Abstract.
A smooth cubic threefold X is unirational: there exists a dominant rational map f : P^3 → X. Clemens and Griffiths proved that a smooth complex cubic X is irrational, i.e. the degree of such f is always > 1. That was the first counter-example to the Lüroth Problem. The difficult part of their proof was to show that the intermediate Jacobian of X (which is an Abelian variety canonically attached to X) is not the Jacobian of a curve. In this talk we will prove that result anew for a generic cubic, by elementary methods: reduction mod p and point counting. This is a joint work with Dimitri Markouchevitch.
January 2018
Antareep
Mandal
HU Berlin
Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
This talk updates our progress towards obtaining the optimal sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. In particular, we study the Maass operators and generalize the relation between the cusp forms and the Maass forms in higher dimensions along with presenting an interesting trick to obtain the heat kernel corresponding to the Maass Laplacians by adapting the 'method of images' used to obtain the heat kernel corresponding to the Laplace-Beltrami operator.
Walter
Gubler
U Regensburg
Non-archimedean Monge-Ampère equations
Abstract.
We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of a metrized line bundle. We prove that the nonarchimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Ampère measure. Such a differentiability formula had been proposed by M. Kontsevich and Y. Tschinkel. In residue characteristic zero, it implies an orthogonality property for non-archimedean plurisubharmonic functions which allows us to drop an algebraicity assumption in a theorem of S. Boucksom, C. Favre and M. Jonsson about the solution to the non-archimedean Monge-Ampère equation. We will also present a similar result in positive equicharacteristic assuming resolution of singularities.
December 2017
Davide Cesare
Veniani
Uni Mainz
Recent advances about lines on quartic surfaces
Abstract.
The number of lines on a smooth complex surface in projective space depends very much on the degree of the surface. Planes and conics contain infinitely many lines and cubics always have exactly 27. As for degree 4, a general quartic surface has no lines, but Schur's quartic contains as many as 64. This is indeed the maximal number, but a correct proof of this fact was only given quite recently. Can a quartic surface carry exactly 63 lines? How many can there be on a quartic which is not smooth, or which is defined over a field of positive characteristic? In the last few years many of these questions have been answered, thanks to the contribution of several mathematicians. I will survey the main results and ideas, culminating in the list of the explicit equations of the ten smooth complex quartics with most lines.
Morten
Risager
U Kopenhagen
Arithmetic statistics of modular symbols
Abstract.
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols. We prove a refined version of this conjecture.
Efthymios
Sofos
MPI Bonn
An Erdős-Kac law for local solubility in families of varieties
Abstract.
A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Daniel Loughran.
November 2017
Ezra
Waxman
U Tel Aviv
Angles of Gaussian primes
Abstract.
Fermat showed that every prime p=1 mod 4 is a sum of two squares: p=a^2+b^2, and hence such a prime gives rise to an angle whose tangent is the ratio b/a. Hecke proved that these Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a conjecture, motivated by a random matrix model, for the variance of Gaussian primes across sectors, and discuss ongoing work about a more refined conjecture that picks up lower-order-terms. I will also introduce a function field model for this problem, which will yield an analogue to Hecke's equidistribution theorem. By applying a recent result of N. Katz concerning the equidistribution of 'super even' characters (the function field analogues to Hecke characters), I will provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime.
Martin
Lüdtke
U Frankfurt
A birational anabelian reconstruction theorem for curves over algebraically closed fields
Abstract.
The question central to birational anabelian geometry is how strongly a field $K$ is determined by its absolute Galois group $G_K$. According to a conjecture of Bogomolov, if $K$ is the function field of a variety of dimension at least 2 over an algebraically closed field, it can by fully recovered from $G_K$. In dimension 1, however, $G_K$ is a profinite free group of rank equal to the cardinality of the base field, containing therefore no information about $K$. We show that a complete reconstruction is possible if one knows in addition how $G_K$ is embedded into the group of field automorphisms fixing only the base field.
October 2017
Enlin
Yang
U Regensburg
Characteristic class and the ε (epsilon) factor of an étale sheaf
Abstract.
In this talk, we will briefly recall the definition and the properties of singular support and characteristic cycle of a constructible complex on a smooth variety. The theory of singular support is developed by Beilinson, which is motivated by the theory of holonomic D-modules. The characteristic cycle is constructed by Saito. In a joint work with Umezaki and Zhao, we prove a conjecture of Kato-Saito on a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field. In our proof, Beilinson and Saito's theory plays an essential role.
July 2017
Charles
Vial
Universität Bielefeld
Distinguished models of intermediate Jacobians
Abstract.
Let X be a smooth projective variety defined over a subfield K of the complex numbers. It is natural to ask whether the complex abelian variety that is the image of the Abel-Jacobi map defined on algebraically trivial cycles admits a model over K. I will show that it admits a unique model making the Abel-Jacobi map equivariant with respect to the action of the automorphism group of the complex numbers fixing K. We offer three applications. First, we show that such a model is a derived-invariant for smooth projective varieties defined over K. Second, we answer a question of Mazur by showing that this model over the base field K is dominated by the Albanese variety of a product of components of the Hilbert scheme of X. Third, we recover a result of Deligne on complete intersections of Hodge level one. This is joint work with Jeff Achter and Yano Casalaina-Martin.
February 2017
Joachim
Mahnkopf
Universität Wien
On local constancy of dimension of slope subspaces of automorphic forms
Abstract.
We prove a higher rank analogoue of a Conjecture of Gouvea-Mazur on local constancy of dimension of slope subspaces of automorphic forms for reductive groups having discrete series. The proof is based on a comparison of Bewersdorff's elementary trace formula for pairs of congruent weights and does not make use of p-adic Banach space methods or rigid analytic geometry.
Bas
Edixhoven
University of Leiden
Group schemes out of birational group laws
Abstract.
In his construction of the jacobian variety of a smooth projective algebraic curve C over a field, Weil first showed that the gth symmetric power of C (with g the genus of C) has a birational group law, and then that this birational group law extends uniquely to a group variety. In the 1960's, Weil's extension result was generalised to schemes by Michael Artin in SGA 3, and used for the construction of reductive group schemes and of Neron models of abelian varieties. At the occasion of the re-edition of SGA 3, I had a look at Artin's article, and it seemed to me that it was better to change the approach to the problem. The main improvement is to construct the group scheme as a sub sheaf of an fppf sheaf of relative birational maps. The fact that relative birational maps admit fppf descent seems to be new. As a consequence, some finiteness conditions in Artin's article are no longer needed. Joint work with Matthieu Romagny: http://arxiv.org/abs/1204.1799 Appeared in Panor. Synthèses, 47, Soc. Math. France, Paris, 2015.
January 2017
Sebastián
Herrero
Chalmers Tekniska Högskola/University of Gothenburg
Asymptotic distribution of Hecke points over Cp
Abstract.
[see here]
Yiannis
Petridis
University College London
Lattice point problems in hyperbolic spaces
Abstract.
[see here]
December 2016
Ania
Otwinowska
U Paris-Sud, Orsay
Around standard conjectures for algebraic cycles
Abstract.
Given a (reasonable) topological space X, one can study its shape by defining a simple invariant attached to X: its cohomology. When moreover X is a complex algebraic variety (i.e. the set of zeroes of a finite collection of complex polynomials) one would like to understand the collection of its algebraic cycles, namely the formal linear combinations of its algebraic subvarieties, and their relations with the cohomology of X. In the 60's Grothendieck proposed a set of simple statements describing some naturally expected relations between algebraic cycles and cohomology: the standard conjectures. In characteristic zero they are implied by the Hodge conjecture. Voisin proved that the following conjecture is a consequence of the standard conjectures: Conjecture N (Voisin): Let X be an algebraic variety. If Z is an algebraic cycle in X whose cohomology class is supported on a closed subvariety Y, then Z is homologically equivalent to a cycle supported on Y. In the first part of this talk, I will present in an elementary way the notion of algebraic cycle, and the above conjectures. In the second part, I will explain a converse to Voisin's result: Theorem (O.) In characteristic 0, Voisin's conjecture N is equivalent to the standard conjectures.
November 2016
Anna
von Pippich
TU Darmstadt
A Rohrlich-type formula for the hyperbolic 3-space
Abstract.
Jensens's formula is a well-known theorem of complex analysis which characterizes, for a given meromorphic function $f$, the value of the integral of $|\log(f(z))|$ along the unit circle in terms of the zeros and poles of $f$ inside this circle. An important theorem of Rohrlich generalizes Jensen's formula for modular functions $f$ with respect to the full modular group, and expresses the integral of $|\log(f(z))|$ over a fundamental domain in terms of special values of Dedekind's Delta function. In this talk, we report on a Rohrlich-type formula for the hyperbolic 3-space.
Remke
Kloosterman
University of Padova
Monomial deformations of Delsarte Hypersurfaces and Arithmetic Mirror Symmetry
Abstract.
In a recent preprint Doran, Kelly, Salerno, Sperber, Voight and Whitcher study for five distinct quartic Delsarte surfaces a one-parameter monomial deformation. Using character sums they find a remarkable similarity between the zeta functions of general members of each of the families. In this talk we present another approach to prove these results. Moreover, we give a necessary and sufficient combinational criterion to check for a pair of monomial deformations of Delsarte hypersurfaces whether their zeta functions are essentially the same or not.
Erik
Visse
University of Leiden
A heuristic for a Manin-type conjecture for K3 surfaces
Abstract.
In the late 1980's Manin came up with a conjecture for the growth of rational points of bounded height of Fano varieties; he predicted what the number of rational points on a suitable open subset of any given such variety should be asymptotically. Since Manin's original paper, many specific cases have been studied giving rise to refinements, proofs, upper and lower bounds, counterexamples and proposed fixes; all still concerning Fano varieties. In my PhD project, I study the same problem for the "next case" in dimension 2: K3 surfaces. Recently I was able to compute heuristics for certain diagonal quartic surfaces that agree with some numerical experiments that were done by my supervisor Ronald van Luijk a few years ago. In the talk I will explain the techniques involved and some problems that need to be overcome in order to formulate a reasonable conjecture.
Antareep
Mandal
HU Berlin
Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
This talk updates our progress towards obtaining the optimal sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. In particular, we study Maass operators and generalize the relation between the cusp forms and the Maass forms in higher dimensions along with presenting an interesting trick to obtain the heat kernel corresponding to the Maass Laplacians by adapting the 'method of images' used to obtain the heat kernel corresponding to the Laplace-Beltrami operator.
July 2016
Martin
Westerholt-Raum
Chalmers University of Technology
Automorphic constituents of tensor products of Harish-Chandra modules
Abstract.
Products of real-analytic automorphic forms in general generate more than one automorphic representation. We study this phenomenon at the infinite place for scalar valued Siegel modular forms of genus 2. It turns out that automorphic constituents of the specific tensor products that we inspect are all holomorphic (limits) of discrete series. This has applications to Poincaré series and harmonic weak Siegel Maaß forms.
Ana Maria
Botero
HU Berlin
Intersection theory of b-divisors on toric varieties
Abstract.
We introduce toric b-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions, toric b-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b-divisor corresponds to the number of lattice points in this convex set and we give a Hilbert-Samuel type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. We further investigate the question of extending these results to arbitrary toroidal varieties. Examples in which such b-divisors naturally appear are invariant metrics on line bundles over toroidal compactifications of mixed Shimura varieties. Indeed, the singularity type which the metric acquires along the boundary can be encoded using toroidal b-divisors.
June 2016
Fabian
Völz
TU Darmstadt
Elliptic and hyperbolic Eisenstein series as theta lifts
Abstract.
Generalising the concept of classical non-holomorphic Eisenstein series associated to cusps, one can define elliptic Eisenstein series associated to points in the upper-half plane, and hyperbolic Eisenstein series associated to geodesics. In my talk I will show that averaged versions of these elliptic and hyperbolic Eisenstein series can be obtained as theta lifts of signature (2,1) of some weighted Poincaré series, which generalizes a classical result in the parabolic case. Moreover, I will show how to realize a distinguished elliptic Eisenstein series as a theta lift of signature (2,2). Finally, if time permits, I will propose some applications of these results.
Jürg
Kramer
HU Berlin
On the volume of the Siegel modular variety
Abstract.
In our talk we provide a short proof of Siegel's formula for the volume of the Siegel modular variety. The proof makes essential use of the Klingen Eisenstein series and is based on a geometric induction argument.
May 2016
Fabien
Pazuki
University of Copenhagen
Bad reduction of curves with CM jacobians
Abstract.
An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in the entry $[I_0-I_0-m]$ in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves. We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over $\overline{\mathbb{Q}}$ with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence such a curve will almost always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit an infinite family of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of $\mathbb{Q}$ of degree 4 that contains $\mathbb{Q}(\sqrt{5})$
Barbara
Jung
HU Berlin
The arithmetic volume of the moduli stack A2
Abstract.
The arithmetic volume of the (compactified) moduli stack An/Z of principally polarized n-dimensional abelian varieties is given by the arithmetic self intersection number of the bundle of Siegel modular forms on An, metrized by the Petersson norm. A generalized intersection theory applicable for this case was developed by Burgos, Kramer and Kühn in 2005. It is conjectured that the above intersection number consists of a sum of special values of the logarithmic derivative of the zeta function. We will present a way to compute the volume of A2, using results from Kudla and Kühn, and discuss how to handle the boundary.
Anton
Mellit
SISSA Trieste
Mixed Hodge structures of character varieties.
Abstract.
The conjecture of Hausel, Letellier and Villegas gives precise predictions for mixed Hodge polynomials of character varieties. In certain specializations this conjecture also computes Hurwitz numbers, Kac's polynomials of quiver varieties, and zeta functions of moduli spaces of Higgs bundles. I will formulate the conjecture, give some examples, and talk about my proof of polynomiality of the generating functions that arise there.
Maryna
Viazovska
HU Berlin
Modular forms and sphere packing
Abstract.
In this talk we will report on our recent result on the sphere packing problem in dimensions 8 and 24. We will explain the linear programming method for sphere packing introduced by N. Elkies and H. Cohn. Also we will present the construction of certificate functions providing the optimal estimate for the sphere packing problem in dimensions 8 and 24.
December 2015
Maryna
Viazovska
HU Berlin
An application of the theory of automorphic forms to discrete geometry
Abstract.
In this talk we will give an overview of classical and recent results on energy optimization problems in discrete geometry. We will focus on the interplay of the theory of automorphic forms and Fourier analysis and their applications to discrete geometry.
Daniel
Loughran
Leibniz Universität Hannover
The Hasse norm principle for abelian extensions
Abstract.
A classical theorem of Hasse states that, for a cyclic extension of number fields L/K, an element of K is a norm from L if and only if it becomes a norm over all completions of K. In this talk, we study the extent to which this "Hasse norm principle" holds for other abelian extensions. Namely, the distribution of abelian extensions of bounded discriminant that fail the Hasse norm principle. This is joint work with Christopher Frei and Rachel Newton.
November 2015
David
Holmes
University of Leiden
Néron models over bases of higher dimension
Abstract.
Néron models for 1-parameter families of abelian varieties were defined and constructed by Néron in the 1960’s, and provide a ‘best possible’ model for the degenerating family. For a degenerating family of abelian varieties over a base scheme of dimension greater than 1, it is much less clear what the ‘best possible' model for the family would be. If one naively extends Néron’s original definition to this setting then these objects fail to exist, even if we allow blowups or alterations of the base space of the family - more precisely, we give a combinatorial characterisation of exactly when such Néron models of jacobians exist. In the case of the jacobian of the universal curve we will describe the minimal base-change required in order that a Néron model exist, giving a possible answer to the shape of the ‘best possible degeneration’.
Anna
von Pippich
TU Darmstadt
On the wave representation of Eisenstein series
Abstract.
Let Γ ⊂ PSL2 (ℝ) be a Fuchsian subgroup of the first kind and let X = Γ\H be the associated finite volume hyperbolic Riemann surface. Eisenstein series attached to parabolic subgroups of Γ play a fundamental role in the theory of automorphic forms on X. Analoguously, one can consider Eisenstein series associated to hyperbolic or elliptic subgroups of Γ. In this talk, we present a unified approach to the construction of these Eisenstein series in terms of the wave kernel. This is joint work with Jay Jorgenson and Lejla Smajlovi'.
Anilatmaja
Aryasomayajula
University of Hyderabad
Heat kernels, Bergman kernels, and estimates of cusp forms
Abstract.
In this talk, we describe a geometric approach to study estimates of cusp forms. The approach relies on the micro-local analysis of the heat kernel and the Bergman kernel. Using which we can derive qualitative estimates of cusp forms of integral weight or half-integral weight associated to arbitrary Fuchsian subgroups and groups commensurable with the Hilbert modular group.
October 2015
Davide
Veniani
Leibniz Universität Hannover
Counting lines on quartic surfaces: New techniques and results
Abstract.
In the last years, two independent research teams (the one based in Ankara, Turkey, the other in Hannover, Germany) have tackled the problem of counting lines on smooth quartic surfaces; the former aimed at a complete classification using computer algebra system GAP, the latter strove for more geometrical insight. The synergy between the two methods has fostered new ideas towards three goals: (1) finding a proof of the fact that the maximal number of lines is 64 which does not involve the flecnodal divisor; (2) proving the uniqueness of the surface with 64 lines with a geometrical approach; (3) adapting the methods to the K3 quartic case. I will report about the state of the art.
July 2015
Ariyan
Javanpeykar
Uni Mainz
Good reduction of complete intersections
Abstract.
In 1983, Faltings proved the Shafarevich conjecture: for a finite set of finite places of a number field K and an integer g>1, the set of isomorphism classes of curves of genus g over K with good reduction outside S is finite. In this talk we shall consider analogues of the Shafarevich conjecture for complete intersections. This is joint work with Daniel Loughran.
June 2015
Antareep
Mandal
HU Berlin
Uniform sup-norm bound for the average over an orthonormal basis of Siegel cusp forms
Abstract.
The average over an orthonormal basis of cusp forms on a given Siegel modular curve can be viewed as the lower part of the discrete spectrum, corresponding to the eigenvalue 0, of the heat kernel associated to the Siegel-Maaß Laplacian. Therefore, one can attempt to use long time asymptotics of this heat kernel to derive an optimum sup-norm bound for the average over an orthonormal basis of cusp forms uniformly in a finite degree cover of the given Siegel modular curve. This method has been proven to work in the one-dimensional case of classical modular forms. In this talk, we present our progress towards a generalization of this method to the higher dimensional case of Siegel modular forms.
Slawomir
Rams
Jagiellonian University Krakow, z.Zt. Leibniz Universität Hannover
On Enriques surfaces with four cusps
Abstract.
One can show that maximal number of A2-configurations on an Enriques surface is four. In my talk I will classify all Enriques surfaces with four A2-configurations. In particular I will show that they form two families in the moduli of Enriques surfaces and I will construct open Enriques surfaces with fundamental groups (Z/3Z)^2 × Z/2Z and Z/6Z, completing the picture of the A2-case and answering a question put by Keum and Zhang. This is joint work with M. Schuett/LUH Hannover.
Valentina
Di Proietto
FU Berlin
On the homotopy exact sequence for the log algebraic fundamental group
Abstract.
There is a strong link between the fundamental group of a variety and the linear differential equations we can define on it. The definition of the fundamental group given in terms of homotopy classes of loops does not generalize easily to algebraic varieties defined over an arbitrary field. But exploiting this link we can give another definition that makes sense in very general contexts: it is called the algebraic fundamental group. We prove the homotopy exact sequence for the algebraic fundamental group for a fibration with singularities with normal crossing and we explain how this gives a monodromy action. This is a joint work with Atsushi Shiho.
Jürg
Kramer
HU Berlin
Degeneration of the hyperbolic heat kernel
Abstract.
In our talk we will investigate the degeneration of the hyperbolic heat kernel and its trace at the cusps of modular curves. This degeneration behavior should be similar to the degeneration of the arithmetic self-intersection number of the corresponding line bundle equipped with a metric that is logarithmically singular at the cusps.
Andriy
Bondarenko
Norwegian University of Science and Technology
On Helson's conjecture
Abstract.
[see here]
May 2015
Wouter
Zomervrucht
Universität Leiden
Bhargava's cube law and cohomology
Abstract.
In his Disquisitiones Arithmeticae, Gauss described a composition law on (equivalence classes of) integral binary quadratic forms of fixed discriminant D. The resulting group is the class group Cl(S), where S is the quadratic algebra of discriminant D. More recently, Bhargava explained Gauss composition as a consequence of a composition law on (equivalence classes of) 2x2x2-cubes of integers. Here one obtains the group Cl(S) x Cl(S). Bhargava's proof is arithmetic. We show how to obtain Bhargava's cube law instead from geometry, with the class groups arising as cohomology. This is work in progress.
Yingkun
Li
TU Darmstadt
Special values of Green function at twisted big CM points
Abstract.
A Green function on an arithmetic variety is a function with logarithmic singularity along an algebraic divisor. Their values at CM points play an important role in the theory of arithmetic intersection. In the case of Hilbert modular surface, one could use Poincare series to explicitly construct the Green function with log singularity along Hirzebruch-Zagier divisors. Its values averaging over Galois orbits of a big CM point are rational numbers and have interesting factorizations. In this talk, we will recall these notions and use harmonic Maass forms of weight one to give a modular interpretation of the values of these Green function at twisted big CM points.
April 2015
Bas
Edixhoven
Universität Leiden
Gästeseminar 'Arithmetische Geometrie' der FU Berlin
February 2015
Henrik
Bachmann
Universität Hamburg
Multiple zeta values and multiple Eisenstein series
Abstract.
In the first half of the talk I will introduce multiple zeta values and discuss their algebraic structure. Multiple zeta values can be seen as a multiple version of the Riemann zeta values appearing in different areas of mathematics and theoretical physics. The product of these real numbers can be expressed in two different ways, the so called stuffle and shuffle product, which yields a large family of linear relations. The second part of the talk is dedicated to multiple Eisenstein series which can be seen as a multiple version of the classical Eisenstein series for the full modular group. By definition the multiple Eisenstein series functions also fulfill the stuffle product. I will explain a recent result which solves the problem of getting also the shuffle product for these functions.
Cecília
Salgado
Universidade Federal do Rio de Janeiro, z.Zt. Leibniz Universität Hannover
Classification of elliptic fibrations on certain K3 surfaces
Abstract.
Let X be an algebraic K3 surface endowed with a non-symplectic involution. We classify all elliptic fibrations on X under some hypothesis on the non-symplectic involution. The idea behind it involves transferring the classification problem to a 'simpler' surface from the geometric point of view. This is work in progress with Alice Garbagnati (Milano).
January 2015
Rafael
von Känel
MPI Bonn
Discriminants and small points of cyclic covers
Abstract.
Let K be a number field. We first consider a generalization of Szpiro's discriminant conjecture to arbitrary smooth, projective and geometrically connected curves X/K of positive genus. Then we present an unconditional exponential version of this conjecture for cyclic covers of the projective line, and we discuss a related work (jointly with A. Javanpeykar) in which we established Szpiro's small points conjecture for cyclic covers. We also plan to explain the proofs. They combine the theory of logarithmic forms with Arakelov theory for arithmetic surfaces.
Amir
Džambić
Kiel
The cohomology of the smallest Hurwitz ball quotient
Abstract.
Recently, M. Stover showed that there exists the unique compact arithmetic 2-dimensional ball quotient of smallest volume. Its smooth Galois coverings, called Hurwitz ball quotients, thus have the maximal automorphism group among the arithmetic ball quotients with the same Euler number. We study the smallest Hurwitz ball quotient and use the knowledge of the automorphisms and the fundamental group to determine its Picard number and the Albanese variety and study some other of its cohomological properties (joint work with Xavier Roulleau).
Kay
Rülling
FU Berlin
Vanishing of the higher direct images of the structure sheaf
Abstract.
Let f: X---> Y be a birational and projective morphism between excellent and regular schemes. Then the higher direct images of the structure sheaf of X under f, R^i f_* O_X, vanish for all positive integers i. In case X and Y are smooth schemes over a field of characteristic zero, this vanishing was proved by Hironaka as a corollary of his proof of the existence of resolutions of singularities. In case X and Y are smooth over a field of positive characteristic the statement was proved by Chatzistamatiou-Rülling in 2011. In this talk I will explain the proof in the general case. This is joint work with Andre Chatzistamatiou.
Jan Hendrik
Bruinier
TU Darmstadt
Classes of Heegner divisors in generalized Jacobians
Abstract.
In parallel to the Gross-Kohnen-Zagier theorem, Zagier proved that the traces of the values of the j-function at CM points are the coefficients of a weakly holomorphic modular form of weight 3/2. Later this result was generalized in different directions and also put in the context of the theta correspondence. We recall these results and report on some newer aspects, which arise from considering classes of Heegner divisors in generalized Jacobians. This is joint work with Y. Li.
Maryna
Viazovska
HU Berlin
Strongly regular graphs from the geometrical point of view
Abstract.
A strongly regular graph with parameters (v, k, l, m) is a k-regular graph in which every pair of adjacent vertices has l common neighbors and every pair of non-adjacent vertices has m common neighbors. In this talk we will give an overview of the theory of these graphs. Also we will report on new non-existance results for strongly regular graphs.
Jürg
Kramer
On the analytic continuation of the heat kernel
Abstract.
In our talk we will present an approach of how to analytically continue the heat kernel associated to the Laplacian of quotient spaces of the hyperbolic plane associated to Fuchsian subgroups of the first kind of PSL(2,R).
Danylo
Radchenko
MPIM Bonn
Higher cross-ratios and functional equations for polylogarithms
Abstract.
A well-known conjecture of Zagier states that the value of a Dedekind zeta function of a number field at an integer m>1 can be expressed in terms of the m-th polylogarithm function. This conjecture remains widely open for all m>3. A general strategy for proving it was outlined by Goncharov, and the main ingredient involves constructing certain higher-dimensional generalizations of the classical cross-ratio. In this talk I will give a general definition of higher cross-ratios, show how they can be used to construct interesting functional equations for polylogarithms, and report on the recent progress towards proving Zagier's conjecture in case m=4.
December 2014
Arne
Smeets
KU Leuven/Orsay
On the insufficiency of the étale Brauer-Manin obstruction
Abstract.
Since Poonen's construction of a variety X defined over a number field k for which X(k) is empty and the étale Brauer-Manin set X(\mathbf{A}_k)^\text{Br,ét} is not, several other examples of smooth, projective varieties have been found for which the étale Brauer-Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using "Poonen's trick", i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this talk, we construct examples for which the Albanese variety is trivial. The new geometric ingredient in our construction is the appearance of Beauville surfaces. Assuming the abc conjecture and using geometric work of Campana on orbifolds, we also prove the existence of an example which is simply connected.
Martin
Bright
Universität Leiden
The Brauer-Manin obstruction and reduction mod p
Abstract.
The Brauer-Manin obstruction plays an important role in the study of rational points on varieties. Indeed, for rational varieties (such as cubic surfaces) it is conjectured that this obstruction determines whether or not a variety has a rational point. I will give a brief introduction to the Brauer-Manin obstruction and then look at recent results relating it to the geometry of the variety at primes of bad reduction.
November 2014
Peter
Bruin
Universität Leiden
Optimal bounds for the difference between the Weil height and the Néron-Tate height for elliptic curves over Qbar
Abstract.
Consider an elliptic curve E over Qbar given by a Weierstraß equation with algebraically integral coefficients. For the purpose of computing Mordell-Weil groups, one would like to bound the difference between the two standard height functions on E(Qbar) (the Weil height and the Néron-Tate height) as sharply as possible. I will describe an algorithm that computes the infimum and the supremum of the difference between these height functions to any desired precision. The main source of difficulties are the Archimedean places; it turns out that these can be treated using the classical Weierstraß elliptic functions.
Benjamin
Göbel
Universität Hamburg
Arithmetic local coordinates and applications to arithmetic self-intersection numbers
Abstract.
In order to calculate the arithmetic self-intersection number of an arithmetic prime divisor on an arithmetic surface, we need to move the prime divisor by the divisor of a rational function. Since there is no canonical choice for the rational function, we may ask whether there is an analytic shadow of the prime divisor that replaces the geometric intersection number at the finite places on the arithmetic surface by an analytic datum on the induced complex manifold. This leads to the definition of an arithmetic local coordinate. In this talk we show that the arithmetic self-intersection number of an arithmetic divisor can be written as a limit formula using an arithmetic local coordinate. We also apply this idea to the intersection theory of H. Gillet and C. Soule and to the generalized intersection theory of J. I. Burgos Gil, J. Kramer and U. Kühn.
Barbara
Jung
HU Berlin
An integral around the toroidal boundary of A2
Abstract.
To obtain the arithmetic degree of the Hodge bundle on a (toroidal) compactification of the Siegel modular variety A2 of degree two, an integral over the regularized star product of Green objects, corresponding to Siegel modular forms, has to be computed. By definition of this product and Stokes’ Theorem, an integral over an ε-tube around the boundary appears. As the integrand degenerates near the boundary, it is not clear that this converges. We will give an explicit description of the geometric situation and show that the integral is in fact going to zero for ε approaching zero.
October 2014
Niels
Lindner
HU Berlin
Bertini theorems for simplicial toric varieties over finite fields
Abstract.
The classical Bertini theorems on generic smoothness and irreducibility do not hold for varieties over finite fields. However, by the work of Poonen and Charles, it is still possible to give a "probability" for smoothness or geometric irreducibility in certain linear systems on subvarieties of projective space over a finite field. Both versions extend to the context of projective simplicial toric varieties. As an application, one finds that "almost all" hypersurfaces in a nice simplicial toric variety are geometrically irreducible and have a finite singular locus, which is small compared to the degree.
July 2014
David
Ouwehand
HU Berlin
Local rigid cohomology of weighted homogeneous singularities
Abstract.
In 2005 Abbott, Kedlaya and Roe have given a method to compute the rigid cohomology of a smooth hypersurface. The goal of this talk is to explain how this method can be applied to the computation of the local rigid cohomology of weighted homogeneous singularities.
Giovanni
de Gaetano
HU Berlin
A metric degeneration approach to a special case of the arithmetic Riemann - Roch theorem
Abstract.
In 2010 G. Freixas proved an arithmetic Riemann – Roch theorem for arbitrary powers of the bundle of cusp forms on modular curves equipped with log-singular metrics; his approach heavily relied on the properties of the moduli space of pointed Riemann surfaces. The goal of the talk is to introduce a different method to prove this theorem, namely the metric degeneration process, which does not require the existence of such a moduli space. Partial results and open problems of such a program will be widely discussed.
Ana Maria
Botero
HU Berlin
The singularities of the invariant metric on the line bundle of Jacobi forms on the universal elliptic curve
Abstract.
A theorem by Mumford implies that an automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety acquiring only logarithmic singularities. This result is the key point of being able to compute arithmetic intersection numbers from these line bundles. It is then natural to ask, whether this result extends to mixed Shimura varieties. In this talk, we examine the case of the sheaf of Jacobi forms on the universal elliptic curve. We see that Mumford's theorem cannot be applied here since a new kind of singularities appear. However, we show that a natural extension in this case is a so called b-divisor. This extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type formula. This is work by J. I. Burgos, J. Kramer and U. Kühn.
June 2014
Slawomir
Cynk
Jagiellonian University Krakow, z.Zt. Universität Mainz
Picard-Fuchs operators for one parameter families of Calabi-Yau threefolds
Abstract.
I will present some methods of computation of Picard-Fuchs operators for one parameter families of Calabi-Yau threefolds. The motivation comes from the Mirror Symmetry and an attempt to classify Calabi-Yau threefolds with the Picard number equal 1. I will discuss examples with special properties: families without points of maximal unipotent monodromy (MUM) and families with several MUM points. This is joint research with D. van Straten (Mainz).
Miguel Grados
Fukuda
HU Berlin
Zeta functions of ray ideal classes and applications to Arakelov geometry
Abstract.
Pursuing asymptotic formulas for the self-intersection number of the relative dualizing sheaf of a modular curve, one encounters zeta functions associated to congruence subgroups. In this talk, we identify such zeta functions and those attached to ray ideal classes of real quadratic fields. The purpose of this identification is to obtain a residue formula at s=1 for the former class of zeta functions. The special case of $\\(Gamma(N)\\)$ is worked out in detail. If time allows, an idelic interpretation of this approach will be presented.
Siddarth
Sankaran
Universität Bonn
Towards an arithmetic Siegel-Weil formula
Abstract.
The Siegel-Weil formula, first discovered by Siegel in 1951, is a classical theorem that equates the integral of a theta function with a special value of an Eisenstein series. A beautiful series of papers by Kudla and Millson from the late 80's casts this theorem a geometric light, in terms of certain 'special cycles' on Hermitian symmetric domains and their quotients. Since then, evidence has emerged for a deeper arithmetic significance of this geometric interpretation; in many cases, there are deep and surprising connections between integral models of these cycles on the one hand, and derivatives of Eisenstein series on the other. In this talk, I will introduce this circle of ideas, which has come to be known as Kudla's programme, and in particular focus on recent developments in the context of unitary Shimura varieties.
May 2014
Zavosh
Amir-Khosravi
HU Berlin
Double coset spaces for the compact unitary groups
Abstract.
I will speak about B. Gross's construction of a definite Shimura curve, its special points and height pairing, and the Gross-Zagier-type formula to which this leads. I will then describe an ongoing project to carry out a similar construction for the compact unitary group U(n), replacing special points by certain special cycles, and defining a corresponding notion of heights. One expects to obtain a generating series that's an automorphic form on U(r,r) with coefficients in Chow groups.
Remke Nanne
Kloosterman
HU Berlin
The Hilbert function of the singular locus of hypersurfaces
Abstract.
Given a zero dimensional scheme $X$ in $P^n$, it is hard to determine if $X$ occurs as the singular locus of a degree d hypersurface. In this talk we use some results on the topology of singular hypersurfaces to obtain restrictions on the Hilbert function of the singular locus of a degree d hypersurface with isolated singularities. This result has several corollaries: It enables us to determine the Mordell-Weil rank of several isotrivial fibrations of abelian varieties. Moreover, it enables us to give constructions of Severi-Enriques varieties with dimension bigger than expected. For the final application let $p_1, ... p_t$ be points in $P^n$ and $ $m an integer. We give a non-trivial lower bound for the degree of a hypersurface in $P^n$ with $m$-fold points at the $p_i$ and no further singularities.
Yuri
Bilu
Université de Bordeaux
Mini Course: The Subspace Theorem in Diophantine Analysis Part II
Abstract.
I will speak on the work of Corvaja, Zannier and others, on applying the Subspace Theorem to integral points on curves and surfaces.
Yuri
Bilu
Université de Bordeaux
Mini Course: The Subspace Theorem in Diophantine Analysis Part I
Abstract.
I will explain the statement of the Subspace Theorem of Schmidt and Schlickewei and will show some of its applications. In particular, I will prove the Adamcszewki-Bugeaud theorem on transcendence of automatic numbers, and the theorem of Corvaja-Zannier-Levin-Autissier on the non-density of integral points on algebraic surfaces.
April 2014
Tonghai
Yang
University of Wisconsin, zzt. MPI Bonn
The story of $j$ and generalizations
Abstract.
The classical modular invariant $j$ has a long history. For example, it was long known that $j(\frac{D +\sqrt D}2)$ is an algebraic integer generating the Hilbert class field of the imaginary quadratic field $Q(\sqrt D)$. As early as 1920's, Berwick observed that the norm of the difference of some singular moduli, like $j(\frac{-163+\sqrt{-163}}2) -j(i)$, although very big, has a very small and interesting prime factorization. In 1985, Gross and Zagier confirmed this guess and gave a beautiful factorization form in general. In 1990s, Borcherds, in attempting to prove the celebrated Moonshine conjecture, discovered and proved a beautiful and surprising product formula for the modular function $j(z_1) -j(z_2)$. His idea of 2nd proof, the regularized theta lifting, can be used to prove Gross-Zagier's formula easily. Moreoever, the method is very soft and can be extended to give direct link between the central derivative of some L-series and the height pairing on some Shimura varieties of orthogonal/unitary type. In this talk, I will mainly focus on the proof of Gross-Zagier formula using regularized theta lifting (less notation and general concepts) after some background review. In the last 20-30 minutes, I will explain the extension.
February 2014
Anna
von Pippich
TU Darmstadt
An arithmetic Riemann-Roch theorem for weighted pointed curves
Abstract.
In this talk, we report on work in progress with G. Freixas generalizing the arithmetic Riemann-Roch theorem for pointed stable curves to the case where the metric is allowed to have conical singularities at the marked points. We will first outline the main ideas of the proof and then focus on some analytical ingredients, e.g. the explicit computation of the regularized determinant for hyperbolic cusps and cones.
Tim
Dokchitser
University of Bristol
L-functions of curves
Abstract.
L-functions of elliptic curves have been studied a lot and their local invariants (local factors, conductors, Tate module, root numbers etc.) are well-understood, both theoretically and computationally. For curves of higher genus the situation is more complicated, and I will report on a joint work in progress with Vladimir Dokchitser that attempts to develop the corresponding theory and classification. Basically, there are two approaches to understand these L-functions, one using regular models and one using semistable models. I will explain what they are and what they can achieve, focussing in particular on hyperelliptic curves over the rationals.
January 2014
James
Stankewicz
University of Copenhagen
Palindromic Properties and Descent Obstructions
Abstract.
For most curves that you might think of, it is possible to find a twist which has a rational point. For the first time we exhibit an infinite family of curves over the rational numbers for which this explicitly does not apply. That is to say that we find Shimura curves C whose lack of rational points is palindromic or preserved by twists. Using this family of curves, we find a related set of twists of Shimura curves which all violate the Hasse Principle. This violation is explicitly given by a descent obstruction.
Frank
Gounelas
HU Berlin
Cohomology of SRC varieties in positive characteristic
Abstract.
This talk will offer two different perspectives on the fact that a separably rationally connected variety in characteristic p has H^1(X, O_X)=0. Over C, this result follows from Hodge theory. The first proof comes from a result of Biswas-dos Santos on triviality of vector bundles on SRC varieties (following from deep recent results of Langer in char p), whereas the second is cohomological (etale and crystalline) in nature. I will introduce some of the necessary background theory and try to include full proofs.
December 2013
Wojciech
Gajda
Adam Mickiewicz University, Poznań
Independence of \\(ell\\)-adic representations
Abstract.
We discuss certain arithmetical properties of Galois representations attached to etale cohomology of algebraic varieties and schemes defined over finitely generated fields of any characteristic. The talk will contain a report on recent joint work with Gebhard Boeckle and Sebastian Petersen.
Aprameyo
Pal
Uni Heidelberg, zzt. HU Berlin
Non-commutative Iwasawa theory and its applications
Abstract.
In Arithmetic Geometry one of the main themes has always been to understand the interplay between analytic invariants and algebraic invariants. One of the most famous example of this interplay is Birch and Swinnerton-Dyer conjecture. Iwasawa theory is one of the important tools which sheds some light on this issue. It provides a crucial link between the characteristic ideal of the Selmer groups (which are defined algebraically) and p-adic L-functions (which are defined analytically). In this talk, I will explain how non-commutative Iwasawa theory fits in the bigger picture. I will explain some results in function field and number field case. If time permits, I will sketch some proofs.
November 2013
José
Burgos
ICMAT, Madrid
Equidistribution of small points on toric varieties
Abstract.
As the culmination of work of many mathematicians, Yuan has obtained a very general equidistribution result for small points on arithmetic varieties. Roughly speaking Yuan's theorem states that given a "very" small generic sequence of points with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to the measure associated to the hermitian line bundle. Here very small means that the height of the points converges to the lower bound of the essential minimum given by Zhang's inequalities. The existence of a very small generic sequence is a strong condition on the arithmetic variety because it implies that the essential minimum attains its lower bound. We will say that a sequence is small if the height of the points converges to the essential minimum. By definition every arithmetic variety contains small generic sequences. We show that for toric line bundles on toric arithmetic varieties Yuan's theorem can be split in two parts: a) Given a small generic sequence of points, with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to a measure. b) If the sequence is very small, the limit measure agrees with the measure associated to the hermitian line bundle.
Remke Nanne
Kloosterman
HU Berlin
Alexander polynomials of curves and Mordell-Weil ranks of Abelian Varieties
Abstract.
Let C={f(z_0,z_1,z_2)=0} be a plane curve with ADE singularities. Let m be a divisor of the degree of f and let H be the hyperelliptic curve [ y^2=x^m+f(s,t,1).] defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of H. For this we use the Alexander polynomial of C. This extends a result by Cogolludo-Augustin and Libgober for the case of where C is a curve with ordinary cusps. In the second part of the talk we sketch how the work of David Ouwehand can be used to obtain a similar result for Jacobians of curves over $\mathbb{F}_q(s,t)$.
Giovanni
De Gaetano
HU Berlin
Towards an arithmetic Riemann Roch theorem for non-compact modular curves
Abstract.
The goal of the talk is to survey the construction and the explicit computation of the Quillen metric on the determinant of cohomology of powers of the Hodge bundle on a modular curve, this corresponds to the definition and the computation of the left hand side of an analogue of Gillet and Soulè's arithmetic Riemann Roch theorem for smooth arithmetic surfaces. In specific we want to describe how the case of the first power of the Hodge bundle is problematic in this situation. If time permits we would like to discuss the right hand side of the formula in the already solved cases, and specify what our task in this context would be.
October 2013
Ana Maria
Botero
HU Berlin
Spherical varieties
Abstract.
First, we will introduce the notion of spherical varieties and discuss important subclasses (horospherical, toroidal) and many examples of them. We present their description by so-called colored fans and, finally, we show how the Tits fibration can be used to understand spherical varieties as T-varieties. Thus, colored fans turn into p-divisors. The latter is recent work by Klaus Altmann, Valentina Kiritchenko and Lars Petersen.
Jürg
Kramer
HU Berlin
Effective bounds for Faltings's delta function
Abstract.
In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces. For a given compact Riemann surface X of genus g, this invariant is roughly given as minus the logarithm of the distance of the point in the moduli space of genus g curves determined by X to its boundary. In our talk, we will first give a formula for Faltings's delta function for compact Riemann surfaces of genus g>1 in purely hyperbolic terms. This formula will then enable us to deduce effective bounds for Faltings's delta function in terms of the smallest non-zero eigenvalue and the shortest closed geodesic of X. If time permits, we will also address a question of Parshin related to bounding the height of rational points on curves defined over number fields.
July 2013
Adrian
Vasiu
U Binghamton
Generalized Serre-Tate Ordinary Theory
Abstract.
In the late 60's, Serre and Tate developed an ordinary theory for abelian varieties over a perfect field of positive characteristic. The ordinary locus corresponds to the generic, dense, open stratum of the Newton polygon stratification of the moduli spaces of principally polarized abelian varieties of a fixed dimension. In this talk, we report on a generalized Serre-Tate ordinary theory that involves Shimura-ordinariness and Uni-ordinariness, in both abstract and geometric contexts. In particular, geometric applications to the study of special fibres of integral canonical models of Shimura varieties of Hodge type and the Shimura-ordinary loci will be presented.
Barbara
Jung
HU Berlin
The star product of Green currents on the Siegel modular variety of degree two
Abstract.
To obtain the arithmetic degree of the Hodge bundle on (a compactification of) the Siegel modular variety A_2 of degree two, the fourfold intersection product of the bundle of modular forms equipped with the Petersson metric has to be computed. This leads to an integral over the regularized star product of corresponding Green currents on A_2. Choosing appropriate currents, we obtain a decomposition in computable integrals over cycles on A_2. We will carry out this decomposition, evaluate the integrals and compare with the expected value.
June 2013
David
Ouwehand
HU Berlin
Rigid cohomology at singular points and the computation of zeta functions
Abstract.
The topic of this talk is the computation of the action of Frobenius on the rigid cohomology of the complement of certain singular projective hypersurfaces over a finite field. Abbott, Kedlaya and Roe have developed a method that solves this problem for smooth hypersurfaces, but for singular hypersurfaces there are still many open questions left. We start by introducing a notion of equivalence of singularities for varieties over finite fields. This notion has the advantage that two equivalent singularities have isomorphic local rigid cohomology. Then we discuss a method for dealing with varieties having weighted homogeneous singularities by using an idea of Dimca. This is where the understanding of the local cohomology at singular points plays a key role.
Lejla
Smajlovic
U Sarajevo
On Maass forms and holomorphic modular forms on certain moonshine groups
Abstract.
We present results on analytical and numerical study of Maass forms and holomorphic modular forms on moonshine groups of level N, where N is a squarefree positive integer. We derive "average" Weyl's law for the distribution of discrete eigenvalues of Maass forms from which we deduce the "classical" Weyl's law. The groups corresponding to levels N=5 and N=6 have the same signature; however, our analysis shows that there are infinitely more cusp forms for N=5. Furthermore, we deduce a Kronecker limit formula for parabolic Eisenstein series and express the "Kronecker limit function" as a geometric mean of product of classical eta functions. We also study holomorphic forms non-vanishing at the cusp and discuss the construction of the j-function(s).
May 2013
Shaoul
Zemel
TU Darmstadt
A Gross-Kohnen-Zagier Type Theorem for Higher-Codimensional Heegner Cycles
Abstract.
The multiplicative Borcherds singular theta lift is a well-known tool for obtaining automorphic forms with known zeros and poles on quotients of orthogonal symmetric spaces. This has been used by Borcherds in order to prove a generalization of the Gross-Kohnen-Zagier Theorem, stating that certain combinations of Heegner points behave, in an appropriate quotient of the Jacobian variety of the modular curve, like the coefficients of a modular form of weight 3/2. The same holds for certain CM (or Heegner) divisors on Shimura curves. The moduli interpretation of Shimura and modular curves yields universal families (Kuga-Sato varieties) over them, as well as variations of Hodge structures coming from these universal families. In these universal families one defines the CM cycles, which are vertical cycles of codimension larger than 1 in the Kuga-Sato variety. We will show how a variant of the additive lift, which was used by Borcherds in order to extend the Shimura correspondence, can be used in order to prove that the (fundamental cohomology classes of) higher codimensional Heegner cycles become, in certain quotient groups, coefficients of modular forms as well. Explicitly, by taking the $m$th symmetric power of the universal family, we obtain a modular form of the desired weight 3/2+m.
Banafsheh
Farang-Hariri
HU Berlin
Local and global theta correspondence
Abstract.
I will start by describing the local theta correspondence also known as Howe correspondence over a local non-archimedean field. This correspondence relates different class of representations of two reductive groups (G,H) called a dual pair by means of the Weil representation. Then I will talk about the global theta correspondence and its relation with the local one. I will also explain the link between global theta correspondence and theta series in the theory of automorphic forms.
Stefan
Keil
HU Berlin
Non-square order Tate-Shafarevich groups of non-simple abelian surfaces over the rationals
Abstract.
For an elliptic curve (over a number field) it is known that the order of its Tate-Shafarevich group is a square, provided it is finite. In higher dimensions this no longer holds true. We will present work in progress on the classification of all occurring non-square parts of orders of Tate-Shafarevich groups of non-simple abelian surfaces over the rationals. We will prove that only finitely many cases can occur. To be precise only the cardinalities k=1,2,3,5,6,7,10,13,14,26 are possible. So far, for all but the last three cases we are able to show that these cases actually do occur by constructing explicit examples.
Philipp
Bannasch
HU Berlin
Thetadivisoren auf elliptischen Modulflächen und ihr Schnittverhalten
April 2013
Giovanni
De Gaetano
HU Berlin
Remarks on Determinants of Laplacians on Riemann Surfaces
Niels
Lindner
HU Berlin
Bertini's theorem for weighted projective space over a finite field
Abstract.
The classical Bertini theorem states that a general hypersurface in complex projective space is smooth. Given a smooth subvariety X of projective space over a finite field, one can actually calculate the fraction of hypersurfaces whose intersection with X is again smooth. This number can be expressed in terms of the Zeta function of X. The question makes also sense when replacing 'projective space' with 'weighted projective space' and 'smooth' with 'quasismooth'. However, the nature of weighted projective space raises some new difficulties.
Jan
Bruinier
TU Darmstadt
Heights of Kudla-Rapoport divisors and derivatives of L-functions
March 2013
Aprameyo
Pal
Univ. Heidelberg
Functional equation of characteristic elements of Abelian varieties over function fields
Abstract.
In this talk we apply methods from the Number field case of Perrin-Riou & Zabradi in the Function field set up. In Zl- and GL2-case (l≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the Main conjectures of Iwasawa Theory. We also prove some parity conjectures in commutative and non-commutative cases. As consequence, we also get results on the growth behaviour of Selmer groups in extension of Function fields.
February 2013
Vladimir
Dokchitser
Cambridge
Parity of ranks of elliptic curves
Abstract.
It is in general very difficult to compute ranks of elliptic curves over number fields, even if equipped with any conjectures that are available. On the other hand, the parity of the rank is (conjecturally) very easy to determine -- it is given as a sum of purely local terms, which have a reasonably simple classification. Since 'odd rank' implies 'non-zero rank' implies 'the curve has infinitely many points', this leads to a number of (conjectural!) arithmetic phenomena. The second part of the talk will concern the 'parity conjecture' - that the parity of the rank that is predicted by the Birch-Swinnerton-Dyer conjecture agrees with the prediction of the Shafarevich-Tate conjecture.
Anil
Aryasomayajula
HU Berlin
Towards an extension of a key identity to Hilbert modular surfaces
Abstract.
In 2006, J. Jorgenson and J. Kramer came up with a beautiful identity relating the canonical and hyperbolic volume forms on a Riemann surface. In this talk, we report the progress of the on-going work in collaboration with Nahid Walji, towards an extension of this formula to Hilbert modular surfaces.
January 2013
Maryna
Viazovska
Univ. Köln
CM Values of Higher Green's Functions
Abstract.
Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, and satisfy the equation Δ f = k(1 - k)f, where Δ is the hyperbolic Laplace operator and k is a positive integer. The significant arithmetic properties of these functions were disclosed in the paper of B. Gross and D. Zagier "Heegner points and derivatives of L-series" (1986). In particular, it was conjectured that higher Green's functions have "algebraic" values at CM points. In this talk we will present a proof of the conjecture for any pair of CM points lying in the same quadratic imaginary field. Moreover, we give an explicit factorization formula for the algebraic number obtained (up to powers of ramified primes).
Jürg
Kramer
HU Berlin
Towards arithmetic intersections on mixed Shimura varieties
Damian
Rössler
Toulouse
A direct proof of the equivariant Gauss-Bonnet formula on abelian schemes
Abstract.
We shall present a proof of the relative equivariant Gauss-Bonnet formula for abelian schemes, which does not rely on the Grothendieck-Riemann-Roch theorem. This proof can be carried through in the arithmetic setting (i.e. in Arakelov theory) and leads to interesting analytic questions.
December 2012
Özlem
Imamoglu
ETH Zürich, z.Zt. HU Berlin
On weakly harmonic Maass forms and their Fourier coefficients
Abstract.
Harmonic Maass forms have been studied extensively in last several years due to their connection to several arithmetic problems. Most notably in the half integral weight case, it can be shown that the Ramanujan's mock theta functions and generating functions of traces of singular moduli can be understood in terms of harmonic Maass forms. A simpler example is provided by the non-holomorphic Eisenstein series of weight 2. In this talk, after giving a short overview of the subject, I will show how to construct a distinguished basis of such forms in the case of weight 2 and study their relation of the regularized inner products of modular functions.
Tony
Varilly-Alvarado
Rice University, currently at EPFL Lausanne
Failure of the Hasse principle on general K3 surfaces
Abstract.
Transcendental elements of the Brauer group of an algebraic variety, i.e., Brauer classes that remain nontrivial after extending the ground field to an algebraic closure, are quite mysterious from an arithmetic point of view. These classes do not arise for curves or surfaces of negative Kodaira dimension. In 1996, Harari constructed the a 3-fold with a transcendental Brauer-Manin obstruction to the Hasse principle. Until recently, his example was the only one of its kind. We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class α that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X,α) is constructed from a double cover of P2 × P2 ramified over a hypersurface of bi-degree (2, 2). This is joint work with Brendan Hassett.
Miguel
Grados
Univ. Peruana de Ciencias Aplicadas
L-series of Elliptic Curves with Complex Multiplication
Abstract.
L-series of elliptic curves are complex functions that carry arithmetical information of the curve. From the analytic point of view, it is desirable that those functions possess additional properties such as Euler product expression, analytic continuation to the entire complex plane and a functional equation. Remarkably, such properties hold for the L-series associated to elliptic curves with complex multiplication. In this talk, we will elaborate on the role played by complex multiplication in the context of L-series. Additionally, since the above properties remind us of the Prime Number Theorem in arithmetic progressions, we will discuss how the analytic methods used in Dirichlet's proof can be adjusted to yield a similar theorem for the L-series of elliptic curves.
November 2012
Jan Steffen
Müller
Univ. Hamburg
A p-adic BSD conjecture for modular abelian varieties
Abstract.
In 1986 Mazur, Tate and Teitelbaum came up with a p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes p of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.
Martin
Raum
ETH Zürich
Computational methods, Jacobi forms, and linear equivalence of special divisors
Abstract.
We will start by briefly discussing the influence of computations on mathematics. Some examples will make clear that computations have led to surprising insight into the area of modular forms, and continue doing so. After this general considerations, we will turn our attention to Jacobi forms. Their definition and their connection with ordinary modular forms will be explained in detail. A new algorithm allows us to compute Fourier expansions of Jacobi forms. We will translate this into information about linear equivalences of special divisors on modular varieties of orthogonal type.
Alina C.
Cojocaru
Univ. of Illinois at Chicago, z.Zt. Univ. Göttingen
Frobenius fields for elliptic curves
Abstract.
Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let πp be the p-Weil root of E and Q(πp) the associated imaginary quadratic field generated by πp. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Q(πp) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. The latter is joint work with Henryk Iwaniec and Nathan Jones.
October 2012
Niels
Lindner
HU Berlin
Cuspidal plane curves and their Alexander polynomials
Abstract.
The Alexander polynomial is a useful invariant of the fundamental group of a curve in the complex projective plane. It is strongly connected with the singularities of the curve. If one restricts to curves whose singular points are either ordinary double points or ordinary cusps, one can use calculations on the Mordell-Weil rank of certain associated elliptic three-folds to obtain a nice description of the Alexander polynomial in terms of syzygies of the ideal of cusps.
Giovanni
De Gaetano
HU Berlin
A regularized determinant for the hyperbolic Laplacian on modular forms
Abstract.
With the goal to generalize the work of T. Hahn, who was able to prove an arithmetic Riemann-Roch type formula for the Hodge bundle on a modular curve with log-singular metric, we define and compute a suitable metric for the determinant bundle of a power of the Hodge bundle. It will be a non-smooth analogue of the Quillen metric used by Gillet and Soulé to prove their arithmetic Riemann-Roch theorem.
Bill
Duke
UCLA
Mock-modular forms of weight one and Galois representations
July 2012
Lars
Kühne
ETH Zürich
An effective result of André-Oort type
Abstract.
"The André-Oort Conjecture (AOC) states that the irreducible components of the Zariski closure of a set of special points in a Shimura variety are special subvarieties. Here, a special variety means an irreducible component of the image of a sub-Shimura variety by a Hecke correspondence. The AOC is an analogue of the classical Manin-Mumford conjecture on the distribution of torsion points in abelian varieties. I will present a rarely known approach to the AOC that goes back to Yves André himself: Before the model-theoretic proofs of the AOC in certain cases by the Pila-Wilkie-Zannier approach, André presented the first non-trivial proof of the AOC in case of a product of two modular curves. In my talk, I discuss results in the style of André's method, allowing to actually compute all special points in a non-special curve of a product of two modular curves and more..."
Barbara
Jung
HU Berlin
Computations towards the arithmetic self intersection number of the bundle of modular forms on A2
Abstract.
To compute the self intersection number of the bundle of modular forms on A2, one has to choose sections tactically, as one has to integrate over forms and subvarieties defined by them. I will present a computation of one of the arising terms via induction A1 → A2.
June 2012
Jürg
Kramer
HU Berlin
Remark on a question of Parshin
Hannah
Enders
HU Berlin
On perfect sheaves of modules
Abstract.
In 1957, Rees defined perfect modules as modules where the homological invariants grade and projective dimension coincide. Starting from this point, we will investigate under which conditions a definition of perfect sheaves of modules is possible. Furthermore, we will show that perfect sheaves of modules inherit some interesting properties of perfect modules.
Wouter
Castryck
Univ. Leuven
Curve gonalities and Newton polygons
Abstract.
Let Delta be a lattice polygon, i.e. the convex hull in R2 of a finite number of points of Z2 (called "lattice points"). Assume that it is not contained in a line. Then it is well-known that a generic Laurent polynomial f(x,y) having Delta as its Newton polygon defines a curve whose genus equals the number of lattice points in the interior of Delta. In this talk we will search for combinatorial interpretations for other discrete invariants, such as the Clifford index, Clifford dimension, and the gonality. The latter is by definition the minimal degree of a morphism to P1.
Andrea
Cattaneo
Univ. Mailand
Elliptic CY threefolds over surfaces
Abstract.
In this talk I want to give a classification of the elliptic CY threefolds we can find in a projective bundle over a base surface B. More in detail, if L is an ample line bundle on B, then I want to give explicit bounds on the pairs (a,b) such that in the bundle P(O + La +Lb) the generic anticanonical variety is smooth. I will also give a detailed description in the case B = P2. As an application of this classification, I will switch to physics and show a nice result concerning string theory, which generalizes a result by Aluffi-Esole.
May 2012
Christian
Liedtke
Univ. Bonn
On the Birational Nature of Lifting
Abstract.
Whenever a variety X lifts from characteristic p to characteristic zero, say over the Witt ring, then many classical results over the complex numbers hold for X, and certain "characteristic p pathologies" cannot occur, simply because one can reduce modulo p (I will discuss this in examples). But then, lifting results are difficult, and generally, varieties do not lift. However, in many situations, it is possible or easier to lift a birational model of X, maybe even one that has "mild" singularities (again, I will give examples). So, a natural question is whether the liftability of such a birational model implies that of our original X. We will show that this completely fails in dimension at least 3, that this question is surprisingly subtle in dimension 2, and that it is trivial in dimension 1.
Anna
von Pippich
HU Berlin
On the spectral zeta function of a hyperbolic cusp or cone
Abstract.
In a joint project with G. Freixas we aim at establishing an arithmetic Riemann-Roch isometry for singular metrics. In this talk we report on an analytic ingredient, namely the computation of the regularized determinant of the hyperbolic Laplacian on a cusp or cone with Dirichlet boundary conditions.
Nuno
Freitas
Univ. Barcelona
The generalized Fermat-type equations x^5+y^5 = 2^z^p or 3^z^p via Q-curves
Abstract.
In order to attack the generalized Fermat equation Ax^p+By^q = Cz^r the modular approach to Diophantine equations that initially led to the proof of Fermat's Last Theorem has been progressively refined. In this talk we will explain how several pieces of the strategy need to be generalized in order to solve equations of the form x^5+y^5 = Cz^p. In particular, we will show how we can use two simultaneous Frey-curves defined over Q(√5) to solve the previous equations for a set o primes with density 3/4.
April 2012
Emmanuel
Ullmo
Université Paris-Sud, Orsay
The hyperbolic Ax-Lindemann conjecture for projective Shimura varieties and some applications to the André-Oort conjecture
Jakob
Scholbach
Univ. Münster
Motives and Arakelov theory
Abstract.
Beginnend mit einer kurzen Einführung in die stabile Homotopiekategorie von Schemata werden wir eine neue Kohomologietheorie namens Arakelov-motivischer Kohomologie vorstellen. Diese kann als Variante (und Verallgemeinerung) von arithmetischen K- und Chow-Gruppen angesehen werden. Wir diskutieren einige Eigenschaften wie den arithmetischen Satz von Riemann-Roch sowie, falls Zeit bleibt, die Beziehung zu speziellen L-Werten.
February 2012
Alice
Garbagnati
Mailand
Symplectic and non symplectic automorphisms of K3 surfaces
Abstract.
Fixed a particular family of K3 surfaces, the automorphisms group of a general member is very often unknown. In order to understand properties of the automorphisms groups of K3 surfaces it seems better to fix a particular group and to find the family of K3 surfaces admitting that group as subgroup of the full automorphisms group. The aim of this talk is to present some results on moduli spaces of K3 surfaces admitting a certain finite group $G$ as subgroup of the group of the automorphisms. We consider both groups acting symplectically on the surface (i.e. preserving the nowhere vanishing holomorphic two form) and group acting purely non symplectically (i.e. there is no elements in the group which preserve the nowhere vanishing holomorphic two form). One of the main results we present is that there exist some pairs of groups $(G,H)$ such that $G$ is a subgroup of $H$ and, under some hypothesis, a K3 surface $X$ admits $G$ as subgroup of the automorphisms group if and only if it admits $H$ as subgroup of the automorphisms group. This phenomenon happens in three distinct situations: both $G$ and $H$ act symplectically on the K3, both $G$ and $H$ act purely non symplectically on the K3, $G$ acts purely non symplectically and $H$ contains elements which are symplectic and elements which are non symplectic. Some of the results presented are obtained in collaboration with Alessandra Sarti.
Özlem
Imamoglu
ETH Zürich
Periods of modular forms
January 2012
Peter
Scholze
Univ. Bonn
Die Hasse-Weil-Zeta-Funktion von Modulkurven
Abstract.
Ein alter Satz besagt, dass die Zeta-Funktion von Modulkurven geschrieben werden kann als Produkt von L-Funktionen von Modulformen und Hecke-Charakteren. Insbesondere folgt, dass sie eine meromorphe Fortsetzung hat und die erwartete Funktionalgleichung erfüllt. Der ursprüngliche Beweis beruht auf Arbeiten von Eichler, Shimura, Langlands, Deligne, und Carayol. Wir erklären die Methode von Langlands, und zeigen, wie diese erweitert werden kann, um an Primstellen schlechter Reduktion einen neuen, vereinfachten Beweis dieses Satzes zu liefern.
Evelina
Viada
Basel
Anomalous Varieties and the effective Mordell-Lang Conjecture
Abstract.
We consider an algebraic variety $V$ embedded in a product of elliptic curves $E^N$. It may happen that components of the intersection of $V$ with a proper algebraic subgroup of $E^N$ have dimension larger than expected. Such components are called $V$ anomalous varieties. The non density of all $V$ anomalous varieties in $V$, for all $V$ not contained in any algebraic subgroup, implies the Mordell-Lang Conjecture. Effective bounds for the height and degree of the maximal $V$ anomalous varieties gives the effective Mordell-Lang Conjecture. We will discuss these implications. We will give some new results for varieties of codimension 2 and some cases of the effective Mordell-Lang Conjecture for curves.
Peter
Toth
HU Berlin
A Geometric Proof of the Tamely Ramified Geometric Abelian Class Field Theory
Abstract.
Unramified geometric abelian class field theory establishes a connection between the Picard group and the abelianized etale fundamental group of a smooth, projective, geometrically irreducible curve over a finite field. We begin with a fairly detailed discussion of the unramified theory concentrating on Deligne's geometric proof. Then we turn to the tamely ramified theory, which transforms the classical situation to the open complement of a finite set of closed points of the curve, establishing a connection between a generalized Picard group and the tame fundamental group of the curve with respect to this finite set of closed points and present a geometric proof for the tamely ramified theory.
Amador
Martin-Pizarro
Lyon, z.Z. HU Berlin
On Schanuel's conjecture and CIT
Abstract.
Schanuel's conjecture states that given n complex elements (x_1,...,x_n) linearly independent over \\(mathbb{Q}\\), then \\mathrm{tr.deg} (x_1,...,x_n, \exp(x_1),...,\exp(x_n) )\geq n. In recent research, Zilber has shown that there is a unique \emph{universal} field (up to isomorphism) of cardinality continuum equipped with a group homomorphism \exp : \\(mathbb{G}_a\\) to \\(mathbb{G}_m\\) where Schanuel's conjecture holds and such that the exponential-closure of a finite set is countable. The question remains whether (\\(mathbb{C}\\),\exp) is that field and in particular whether the key obstacle is Schanuel's conjecture itself. In order to prove some of the results, Zilber introduced a weak version of the \emph{Conjecture of Intersection with Tori}, which states that there is only finitely many cosets of tori (uniformely) for a given closed subvariety V of \\mathbb{G}_m\\ describing all possible atypical intersections of V with any proper torus. In this talk, we will present some of the ideas in Zilber's work as well as an approach to weak CIT in positive characteristic by introducing Hasse-Schmidt iterative derivations in a separably closed field and relate the above to the existence of infinite Mersenne primes.
Gérard
Freixas i Montplet
Univ. Paris 6
On Faltings theorem for abelian schemes over arithmetic surfaces
Abstract.
In this talk I will review the statement of the Tate conjecture for abelian varieties. In the number field case this is the celebrated theorem of Faltings. Faltings himself showed how to reduce the case of abelian schemes over higher dimensional bases to abelian schemes over a number field. His proof uses Hodge theory. We will give another approach that avoids Hodge theory but relies on higher dimensional Arakelov geometry. This will be the occasion to state some generalizations of diophantine statements in this theory. The contents of the talk will be based on joint work with Jean-Benoit Bost.
December 2011
Jeng-Daw
Yu
Taiwan National Univ.
The irregular Hodge filtration
Abstract.
This is a report of work in progress with Esnault. Motivated by various cohomology theories of exponential sums over finite fields, we propose a Hodge-type filtration on the cohomology attached to an exponentially twisted de Rham complex over a complex smooth (quasi-projective) variety. I will indicate some good properties of this filtration and list some natural questions.
Matthias
Schütt
Univ. Hannover
Modularity of Maschke's octic and Calabi-Yau threefold
Abstract.
Maschke's octic surface is the unique invariant of a particular group of size 11520 acting on projective fourspace. Recently Bini and van Geemen studied this surface and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by the Heisenberg group. In particular they computed a decomposition of the cohomology in terms of the group and conjectured its modularity. We will sketch how to actually prove this for all three varieties. The proofs rely on automorphisms of the varieties and in one case on isogenies of K3 surfaces.
November 2011
Stefan
Keil
HU Berlin
Shafarevich-Tate groups of non-square order
Abstract.
The order of the Shafarevich-Tate group (=sha) of an elliptic curve over a number field is, if it is finite, a square number. For abelian varieties in higher dimensions this is no longer the case. However, for principally polarized abelian varieties over a number field this is almost true, since then the order of sha is a square or twice a square. The only known example of an abelian surface over QQ having order of sha not equal to a square or twice a square has order of sha equal to three times a square. We will explain how to construct abelian surfaces over QQ having order of sha equal to five or seven times a square.
Jürg
Kramer
HU Berlin
A note on scattering matrices
Philipp
Habegger
Univ. Frankfurt am Main
Beyond the André-Oort Conjecture (joint with Jonathan Pila)
Abstract.
An isomorphism class of elliptic curves defined over C can be identified with a complex number by virtue of Klein's j-function. The so-called singular j-invariants are particularly interesting from an arithmetic point of view. These come from elliptic curves with complex multiplication. A particular case of the André-Oort Conjecture describes the distribution of points on subvarieties of affine n-space whose coordinates are singular j-invariants. Here the conjecture is known due to work of André, Edixhoven, and Pila. Pink's more general conjecture describes points on subvarieties that satisfy moduli theoretic properties which are generally weaker than asking for complex multiplication. This includes examples such as points in affine n-space whose coordinates are pairwise isogenous elliptic curves. I will present progress into the direction of Pink's Conjecture. Our method of proof relies on the theory of o-minimal structures which has its origins in model theory; the general strategy was developed originally by Zannier. In the talk I will explain what an o-minimal structure is and how it interacts with arithmetic components of our proof.
Ulrich
Terstiege
Univ. Duisburg-Essen
Intersections of special cycles on unitary Rapoport-Zink spaces of signature (1,n-1)
Abstract.
We discuss results on intersections of special cycles on unitary Rapoport-Zink spaces that can be applied to the conjectures of Kudla and Rapoport on intersections of special cycles on unitary Shimura varieties and to the arithmetic fundamental lemma conjecture of W. Zhang.
Giovanni
De Gaetano
HU Berlin
On the fundamental group of the affine line in positive characteristic
Abstract.
In the first part of the talk we introduce the notion of fundamental group of a scheme by a pure categorical point of view. This construction is a generalization of the fundamental group of a topological space and of the Galois group of a field. Then, using a little étale topology, we move the notions of 'loop' and 'neighborhood' from the topological to the arithmetic context. In the last part we see what kind of coverings of the affine line arise when we allow a certain ramification index to be divisible by the characteristic of the field, and we outline a strategy for the solution of the problem. Eventually we will understand that the affine line, in positive characteristic, is very far from being simply connected.
October 2011
Remke N.
Kloosterman
HU Berlin
Zariski triples and equisingular deformations of cuspidal curves
Abstract.
In this talk we construct a Zariski triple, i.e., three plane curves $C_1,C_2,C_3$ of the same degree, with the same number and type of singularities such that the fundamental groups of $\mathbb{P}^2\setminus C_i$ are pairwise non-isomorphic. This is done by calculating the Alexander polynomial of $C_i$. We use this example of a Zariski triple to construct families of singular plane curves such that their equisingular deformation space has larger dimension than expected. In the end we show that if $C$ is a curve of degree at least 13 with non-constant Alexander polynomial then the tangent space of the equisingular deformation space has larger dimension than expected.
July 2011
Doug
Ulmer
Georgia Inst. of Technology
Arithmetic of the Legendre curve in a Kummer tower
Abstract.
Let k be a finite field of odd characteristic, K=k(t), and K_d=k(t^{1/d}). We consider the arithmetic of the Legendre elliptic curve E: y^2=x(x-1)(x-t) over the fields K_d. A remarkable elementary construction gives many points on E over K_d for suitable values of d. Less elementary considerations lead to interesting problems and results on the full Mordell-Weil group E(K_d), on heights, and on the Tate-Shafarevich group of E over K_d.
Bjorn
Poonen
MIT
Random maximal isotropic subspaces and Selmer groups
Abstract.
We show that the p-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over F_p. By modeling this intersection as the intersection of a random maximal isotropic subspace with a fixed compact open maximal isotropic subspace, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. The random model is consistent with Delaunay's heuristics for Sha[p], and predicts that the average rank of elliptic curves is at most 1/2. This is joint work with Eric Rains.
June 2011
Christine
Robinson
Univ. of Illinois at Chicago
On a converse theorem and a Saito-Kurokawa lift for Siegel wave forms
George
Walker
Bristol
Zeta functions of plane curves
Abstract.
The problem of computing zeta functions of varieties over finite fields has received considerable interest in recent years, particularly for the case of curves. I shall outline a new algorithm, based on $p$-adic cohomology and the work of Lauder, which may be applied to a wide range of classes of smooth curves. In particular, it matches in complexity Kedlaya's algorithm for hyperelliptic curves and its generalizations to superelliptic curves. I shall focus on its application to smooth plane curves, where the algorithm improves on the complexity of the previously best known algorithm almost by a factor of $g$, the genus.
Ananyo
Dan
HU Berlin
Noether-Lefschetz locus on projective hypersurfaces
Abstract.
Noether's theorem states that a generic degree $d$ smooth hypersurface in $ ackslashegin{P}^3ackslashegin{end}$ has picard number 1. We define the Noether-Lefschetz to be the smooth hypersurfaces in $ackslashegin{P}^3ackslashegin{end}$ of degree $d$. In my talk we study the geometry of this locus including when is an irreducible component of this locus non-reduced.
May 2011
Gérard
Freixas i Montplet
Univ. Paris 6
Arithmetic Riemann-Roch theorem and Jacquet-Langlands correspondence
Abstract.
In this talk we will review the arithmetic Riemann-Roch theorem for pointed curves and we will show how to combine it with the Jacquet-Langlands correspondence, in order to get equalities between certain arithmetic self-intersection numbers on modular and Shimura curves.
Aaron
Greicius
HU Berlin
Images of Galois representations attached to l-Tate modules
Abstract.
Let A/K be a principally polarized abelian variety with trivial endomorphism ring. In many cases (for example, when dim(A) is 2,6 or odd), the various l-adic Galois representations to GSp associated to the l-Tate modules of A/K are surjective for all but finitely many primes l. In such situations one hopes to find an algorithm for finding, or at least bounding, the finite set of exceptional primes where the Galois representation fails to be surjective. We will consider existing algorithms for elliptic curves and abelian surfaces over *Q* and endeavor to extend these to more general number fields. Time permitting, we will also examine the situation when the endomorphism ring of A is larger than *Z*.
Vincenz
Busch
Univ. Hamburg
Beilinson's conjecture for K_2 of a superelliptic curve
Abstract.
The Beilinson conjecture generalize and unify multiple theorems and conjectures in arithmetic geometry, e.g. the class number formula and the BSD conjecture. In the case of K_2 of algebraic curves the effects of the Beilinson conjectures can actually be observed in concrete calculations. In this talk, we will cover an example of such calculations in the case of a superelliptic curve.
Lenny
Taelman
Univ. Leiden
Special values in characteristic p
Abstract.
I will present a theorem which is a characteristic-p-valued function field analogue of the class number formula and the Birch and Swinnerton-Dyer conjecture. In these special value formulas the multiplicative group (for the CNF) and elliptic curves (for BSD) are replaced by Drinfeld modules. Reference: [http://arxiv.org/abs/1004.4304].
Dennis
Eriksson
Univ. Göteborg
Multiplicities of discriminants
Abstract.
I will discuss some recent formulas for multiplicities of discriminants of polynomials, generalising in one direction those of Ogg's formula for the discriminant of elliptic curves. In the particular case of discriminants of planar curves we obtain more precise information, and we can relate it to finite contributions of Arakelov intersection numbers.
April 2011
Hartwig
Mayer
HU Berlin
On Zhang's admissible intersection theory
Abstract.
In 1993, S.-W. Zhang introduced (based on previous work of T. Chinburg and R. Rumely) an intersection theory for smooth, irreducible curves $X/K$ over a local field $K$ by defining a potential theory on the dual of the reduction graph $R(X)$ of $X$. This theory was used to give first answers to the Bogomolov conjecture. In the first part of this talk, we present Zhang's intersection theory and its arithmetic implication in case of a modular curve. In the second part, we present recent developments in extending Zhang's potential theory to Berkovich curves, and end with some open questions concerning a higher-dimensional analogue, which would be of great interest in the context of Arakelov theory.
Anil
Aryasomayajula
HU Berlin
Estimates of the Canonical Green's Function
Abstract.
In this talk, I will try to present the estimates obtained for the canonical Green's function associated to the canonical metric, on non-compact finite volume hyperbolic Riemann surfaces of genus g > 1, in terms of invariants from hyperbolic geometry.
March 2011
Alex
Bartel
POSTECH, Pohang Univ., Korea
Some applications of integral group representations in number theory
Abstract.
Certain quotients of regulators of number fields or of abelian varieties can be interpreted as purely representation theoretic invariants. I will introduce a technique for analysing such invariants and will apply it to some questions on the arithmetic of number fields and of elliptic curves. One of the main results will be a curious identity linking the fine integral Galois module structure of certain units of number fields and of Mordell-Weil groups to sizes of class groups and of Tate-Shafarevich groups, respectively. There will be plenty of examples along the way, and also some mystery and food for future thought. The talk will be accessible to graduate students with a very modest background in number theory and ordinary representation theory.
February 2011
Anna-Maria
von Pippich
HU Berlin
The Weierstrass $\wp$-function and transcendence questions
Lejla
Smajlović
Univ. of Sarajevo
On the distribution of the zeros of the derivative of the Selberg zeta function
Jürg
Kramer
HU Berlin
On a result of Hoffstein-Lockhart
Abstract.
Information not provided.
January 2011
Barbara
Jung
HU Berlin
On the volume formula of the fundamental domain of the Siegel modular group
Jay
Jorgenson
City College New York/BMS
Progress in bounds for Fourier coefficients of modular forms
Jens
Funke
Univ. of Durham/BMS
On a theorem of Hirzebruch and Zagier
December 2010
Brendan
Creutz
Univ. Bayreuth
Large Tate-Shafarevich Groups
Abstract.
For an abelian variety A over a number field k, the Tate-Shafarevich group of A/k parameterizes principal homogeneous spaces for A/k which have points over every completion of k. It is conjectured that this group is finite. However, there are several results in the literature which show that this group can be arbitrarily large. We will discuss some of these and show, in particular, that the p-torsion in the Tate-Shafarevich group of any principally polarized abelian variety over k is unbounded as one ranges over extensions of k of degree O(p).
Jose Ignacio
Cogolludo
Zaragoza
On the connection between the topology of plane curves, the position of their singular points, and elliptic threefolds
November 2010
Luis
Dieulefait
Universitat de Barcelona
Modularity by propagation: Serre's conjecture and non-solvable base change for GL(2)
Remke
Kloosterman
HU Berlin
Mordell-Weil groups, Syzygies and Brill-Noether theory
Abstract.
For a particular class of elliptic threefolds with base P^2 we discuss the relation between the rank of the Mordell-Weil group and the syzygies of the singular locus of the discriminant curve of the elliptic fibration. From this we deduce that for each high rank elliptic threefold we find examples of triples (g,k,n) such that the locus {[C] M_g | C admits a g^2_{6k} and the image of C has at least n cusps} has much bigger dimension than expected.
Stefan
Keil
HU Berlin
The Sato-Tate Conjecture and the L-Function Method
Abstract.
In the last years Tayler et al. showed that the m-th symmetric power of the L-functions of l-adic Galois-representations of an elliptic curve over QQ are potential automorphic. Therefore they fulfill certain meromorphic properties which are sufficient to use the so called L-function method to prove the Sato-Tate Conjecture (over QQ). We will talk in general about equidistribution and the L-function method and its application (e.g. the prove of Chebotarev's density theorem). We will explain why one can apply the L-function method for the Sato-Tate Conjecture. If there is time we will discuss the exceptional case of CM-curves.
Anilatmaja
Aryasomayajula
HU Berlin
Bounds on canonical Green's function
October 2010
Nahid
Walji
HU Berlin
The Lang Trotter conjecture on average and congruence class bias
July 2010
Morten
Risager
Univ. Kopenhagen
Non-vanishing of Fourier coefficients, Poincaré series, and central values of L-functions.
Abstract.
We discuss Fourier coefficients of modular forms at cusps and non-cuspidal values. We show, without to much effort, that 'generically' these coefficients are all non-vanishing. Yet it is highly non-trivial to prove that for a specific point z_0 the coefficients are non-vanishing. In the simplest case of the discriminant function the non-vanishing of Fourier coefficients at infinity is an old conjecture of Lehmer's. We show how the non-cuspidal analogue of this conjecture is true for certain CM-points. We then discuss how this has applications to non-vanishing of certain Poincaré series and to non-vanishing of certain central values of L-functions. This is joint work with Cormac O'Sullivan.
June 2010
Gérard
Freixas
Univ. Paris 6
Generalizing analytic torsion
Frank
Monheim
Univ Tübingen
Iterierte Integrale automorpher Formen
Andreas
Stein
Univ. Oldenburg
Elliptic Curves and Cryptography - Some (new) attacks to the elliptic curve discrete logarithm problem
Abstract.
In recent years, elliptic curves have become objects of intense investigation because of their significance to public-key cryptography. The major advantage of ECC is that the cryptographic security is believed to grow exponentially with the length of the input parameters. This implies short parameters, short digital signatures, and fast computations. We provide a survey of elliptic curves over finite fields and their interactions with algorithmic number theory. Our main focus will be the discussion of various interesting attacks to the so-called elliptic curve discrete logarithm problem (ECDLP) and their mathematical background as well as their important impact on public-key cryptography. For several attacks, results on algebraic curves, especially hyperelliptic curves, are needed.
Stefan
Müller-Stach
Univ. Mainz
Numerical characterizations of Shimura subvarieties
May 2010
Hartwig
Mayer
HU Berlin
Arakelov theory on the modular curve X_1(N) (Part II)
Hartwig
Mayer
HU Berlin
Arakelov Theory on X_1(N)
Aaron
Greicius
HU Berlin
Abelian varieties with large adelic image of Galois
Moritz
Minzlaff
TU Berlin
p-adic Cohomology of Curves and the Calculation of Zeta Functions
April 2010
Alessandro
Verra
Univ. Roma 3
K3 surfaces and some moduli spaces related to curves
Abstract.
Information for the abstract is not provided in the input.
Anna
von Pippich
HU Berlin
Kronecker Limit Formula
February 2010
Chad
Schoen
Duke University, z.Zt. MPIM Bonn
A numerical test of the generalized Birch and Swinnerton-Dyer Conjecture
Abstract.
The generalized Birch and Swinnerton-Dyer Conjecture in its crudest form asserts that the rank of a Chow group is equal to the order of vanishing of an L-function at the center of the critical strip. We discuss joint work with Jaap Top and Joe Buhler in which the conjecture was put to a modest test.
Barbara
Jung
Univ. Mainz
A version of Luna's theorem for symplectic varieties
Abstract.
In 1973, Domingo Luna proved the existence of an etale slice for certain group actions on varieties: He showed that the algebraic quotient X//G of a variety X by a reductive group G can etale locally be described by the action of the stabilizer G_x of a point x on a subvariety S of X. A symplectic variety carries the additional structure of a non degenerate 2-form. To obtain a suitable quotient in the category of symplectic varieties, one has to pass to a certain subvariety Z_X first before taking the algebraic quotient. For this symplectic quotient, Luna's theorem had to be modified in such a way that we found a symplectic subvariety S of X such that there is an etale morphism from the symplectic quotient Z_S//G_x of S by G_x to the symplectic quotient ZX//G of X by G.
January 2010
Stephen
Meagher
Univ. Freiburg
Equations defining isogeny classes of ordinary Abelian varieties
Abstract.
The moduli space of Abelian varieties can be described in terms of equations which derive from relations holding between the theta null values of Abelian varieties. In characteristic p one can look at the action of the Frobenius and Vershiebung morphisms on these theta nulls when the variety is ordinary. From the work of Tate it is known that the characteristic polynomial of an iterate of the Frobenius morphism describes the isogeny type of an Abelian variety. Using this we describe relations between the level p theta nulls in an isogeny class of ordinary Abelian varieties. This is joint work with Robert Carls.
Fredrik
Strömberg
TU Darmstadt
On newforms and multiplicity of the spectrum for Gamma_0(9)
Abstract.
Through numerical investigations it was discovered (by two independent groups of researchers) some years ago that the spectrum of the Laplace-Beltrami operator on $\Gamma_0(9)$ possessed a peculiar feature. Namely, that there did not seem to be any eigenvalues with multiplicity one, that is, even the newforms (in the sense of Atkin-Lehner extended to non-holomorphic forms) appeared to be degenerate. The underlying phenomenon providing this multiplicity became only recently clear. I will give an overview of the proof of a precise version of this observation. The proof involves a mix of spectral theory, in terms of the Selberg trace formula, together with Hecke operators and symmetries of a (slightly overlooked) congruence group of level 3, $\Gamma^3$. I will also discuss how this relates to some of the multiplicity one results for automorphic representations.
Ricardo
Menares
EPFL Lausanne
Functoriality in the Arakelov geometry of arithmetic surfaces. Applications to Hecke operators on modular curves
Abstract.
In the context of arithmetic surfaces, J.-B. Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pullback and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are selfadjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due to U. Kuehn we compute these invariants in terms of special values of L series.
Ananyo
Dan
HU Berlin
Classifying the biggest components of the Noether-Lefschetz locus
Abstract.
Information not provided.
December 2009
Bisi
Agboola
UCSB z.Zt. HU Berlin
Restricted Selmer groups and special values of p-adic L-functions
Abstract.
I shall discuss conjectures of Birch and Swinnerton-Dyer type involving special values of the Katz 2-variable p-adic L-function that lie outsied the range of p-adic interpolation.
November 2009
Amaury
Thuillier
Lyon 1
Non-Archimedean analytic geometry and Arakelov theory
Abstract.
Arakelov geometry on an algebraic variety X over Q usually combines intersection theory on a model of X over the integers with analysis on the corresponding complex analytic variety. Relying on Berkovich's approach of p-adic analytic geometry, it is possible to replace integral models by (real) analysis at each non-Archimedean place of Q. This extension of Arakelov geometry is particulary suitable for the study of canonical heights, or to formulate and prove p-adic equidistribution theorems. I will explain this without assuming a specific knowledge of Berkovich theory.
Peter
Bruin
Univ. Leiden
Arakelov theory and height bounds
Abstract.
In the work of Bas Edixhoven and others on computing two-dimensional Galois representations associated to modular forms over finite fields, part of the output of the algorithm is a certain polynomial with rational coefficients that is approximated numerically. Arakelov's intersection theory on arithmetic surfaces is applied to modular curves in order to bound the heights of the coefficients of this polynomial. I will explain the connection between Arakelov theory and heights, indicate what quantities need to be estimated, and give methods for doing this that lead to explicit height bounds.
Mark
Blume
HU Berlin
Losev-Manin moduli spaces and toric varieties associated with root systems
Judith
Ludwig
Univ. Tübingen
Parabolic cohomology and rational periods
October 2009
Christian
Schön
HU Berlin
Some \\Gamma_1(N) modular forms and their connection to the Weierstrass \\wp-function
Kira
Samol
Univ. Mainz
Dwork congruences and reflexive polytopes
July 2009
H.
Shiga
Chiba University
and
J.
Estrada Sarlabous
Havana
A Jacobi type formula in two variables with application to a new AGM; Non-hyperelliptic curves of genus 3 and the DLP
Abstract.
The index calculus algorithm of Gaudry, Thomé, Thériault, and Diem makes it possible to solve the Discrete Logarithm Problem (DLP) in the Jacobian varieties of hyperelliptic curves of genus 3 over F_q in O(q^{4/3}) group operations. On the other hand, applied to Jacobian varieties of non-hyperelliptic curves of genus 3 over F_q, the index calculus algorithm of Diem requires only O(q) group operations to solve the DLP. This vulnerability to faster index calculus attacks of the non-hyperelliptic curves of genus 3 has discouraged the use of Jacobian varieties of non-hyperelliptic curves of genus 3 as a basis of DLP-based cryptosystems. A recent work of B. Smith introduces the idea of exploiting this vulnerability to faster index calculus attacks of the non-hyperelliptic curves of genus 3 to discard a non-negligible subset of hyperelliptic curves of genus 3 over F_q. I would like to expose some highlights of this approach of B. Smith (the mathematical ingredients are nice: isogenies of Jacobian varieties, Recilla's trigonal construction, etc.) and to discuss the interest of studying non-hyperelliptic curves of genus 3 in the context of DLP-based cryptosystems.
Anna
von Pippich
HU Berlin
The arithmetic of elliptic Eisenstein series
June 2009
Remke Nanne
Kloosterman
HU Berlin
Average rank of elliptic n-folds
Abstract.
For elliptic curves over number fields it is conjectured that the half the curves have rank 1 and half the curves have rank 0. Similarly, if C/F_q is a curve then it is conjectured the half the elliptic curves over F_q(C) have rank 0 and half the curves have rank 1. In this talk we show that the situation is different if one considers elliptic curves over F_q(V), with dim(V)>1.
Ronald
van Luijk
Leiden
Explicit two-coverings of Jacobians of genus two
May 2009
Anil
Aryasomayajula
HU Berlin
A fundamental identity of metrics on hyperbolic Riemann surfaces of finite volume
Shabnam
Akhtari
MPI Bonn
Representation of Integers by Binary Forms
Abstract.
Suppose F(x,y) is an irreducible binary form with integral coefficients, degree n >= 3 and discriminant D_F \neq 0. Let h be an integer. The equation F(x,y)=h has finitely many solutions in integers x and y. I shall discuss some different approaches to the problem of counting the number of integral solutions to such equations. I will give upper bounds upon the number of solutions to the Thue equation F(x,y)= h. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential rule in this study.
Anna
Posingies
HU Berlin
A Kronecker Limit Formula for Fermat Curves
April 2009
Hartwig
Mayer
HU Berlin
Arithmetic self-intersection number of the dualizing sheaf on modular curves
February 2009
Aaron
Greicius
HU Berlin
Elliptic curves with surjective adelic Galois representation
Abstract.
Let E/K be an elliptic curve over a number field K. Let G_K be the absolute Galois group of K. The action of G_K on the torsion points of E gives rise to an \emph{adelic} Galois representation $$\rho:G_K\rightarrow Aut(E_{tor}(\overline{K}))\simeq \GL_2(\hat{\mathbb Z}).$$ In 1972 Serre proved that the image of \rho is open, hence of finite index, when E/K is non-CM. The question naturally arises then whether this index is ever equal to 1. In other words, is it possible for \rho to be surjective? By examining the maximal closed subgroups of $GL_2(\hat{\mathbbZ})$, we come up with simple necessary and sufficient conditions for this to be the case. This allows us to find examples of number fields $K\ne\mathbb{Q}$ and elliptic curves E/K for which \rho is indeed surjective.
Amir
Dzambic
Univ. Frankfurt am Main
Hyperbolic 3-manifolds with maximal automorphism group
January 2009
Jean-Benoit
Bost
Univ. Paris-Sud, Orsay
Unitary integrable connections defined over number fields
Jay
Jorgenson
CCNY
Sup-norm bounds for weight k automorphic forms
Rolf-Peter
Holzapfel
HU Berlin
Picard modular del Pezzo surfaces, II
K.
Künnemann
Univ. Regensburg
Line bundles with connections on projective varieties over function fields and number fields
Abstract.
We report about joint work with with Jean-Benoit Bost (Orsay). Consider a hermitian line bundle on a smooth, projective variety over a number field. The arithmetic Atiyah class of the hermitian line vanishes by definition if and only if the unitary connection on the hermitian line bundle is already defined over the number field. We show that this can happen only if the class of the line bundle is torsion. This problem may be translated into a concrete problem of diophantine geometry, concerning rational points of the universal vector extension of the Picard variety. We investigate this problem, which was already considered and solved in some cases by Bertrand, by using a classical transcendence result of Schneider-Lang. We also consider a geometric analog of our arithmetic situation, namely a smooth, projective variety which is fibered on a curve defined over some field of characteristic zero. To any line bundle on the variety is attached its Atiyah class relative to the base curve. We describe precisely when this relative class vanishes. In particular, when the fixed part of the relative Picard variety is trivial, this holds only when the restriction of the line bundle to the generic fiber of the fibration is a torsion line bundle.
December 2008
Philipp
Habegger
ETH Zürich
Points with multiplicative dependent coordinates on curves
Abstract.
The Mordell Conjecture states that a smooth projective curve of genus at least 2 has only finitely many points defined over a number field. This is now a theorem of Faltings; another proof using completely different ideas was found by Vojta. His approach can be encapsulated neatly in a single height inequality. Remond found a more uniform version of this inequality and also applications to abelian varieties in the context of the Zilber-Pink Conjecture governing the intersection of a variety with the union of certain algebraic subgroups. Maurin later built up on Remond's result in the toric case and showed that the curve contained in the algebraic torus of dimension 6 parametrized by (2,3,5,t,1-t,1+t) contains only finitely many points whose coordinates satisfy two independent multiplicative relations. Based ultimately on Vojta's method, Maurin's Theorem is ineffective: it does not allow us to determine the t's. We propose a different approach which circumvents Vojta's method and which is in principle effective in the toric setting.
Rolf-Peter
Holzapfel
HU Berlin
Picard modular del Pezzo surfaces
November 2008
Anilatmaja
Aryasomayajula
HU Berlin
Hyperbolic and canonical metrics
Maria
Petkova
HU Berlin
Shimura curve computation
Anna
von Pippich
HU Berlin
Elliptic and hyperbolic degeneration
October 2008
Jürg
Kramer
HU Berlin
Estimating Green's Functions I
July 2008
Andrea
Miller
Harvard Univ.
Chow-Künneth decompositions for some mixed Shimura varieties
Laura
Hinsch
Univ. Göttingen
Projective spaces and sheaves of modules on F_1
Riccardo
Salvati Manni
Universita La Sapienza, Roma I
The loci of abelian varieties with singular points of order two on the theta divisor
June 2008
Daniel
Garbin
CUNY Graduate Center, New York
Spectral convergence of elliptically degenerated Riemann surfaces
Tobias
Hahn
ETH Zürich
Arithmetic Riemann Roch theorem for modular curves
May 2008
Bernhard
Heim
MPI Bonn
On the Modularity of the GL_2-twisted Spinor L-function
Abstract.
There are famous theorems on the modularity of Dirichlet series with Euler product attached to geometric or arithmetic objects. There is Hecke's converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat's Last Theorem to name a few. In this talk in the spirit of the Langlands philosophy we raise the question on the modularity of the GL_2-twisted spinor L-function L(s,G imes h) related to automorphic forms G, h on the symplectic group with similitudes GSp(4) of degree 2 and GL_2. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insides into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan's work on the modularity of the Rankin-Selberg L-series.
Vladimir
Berkovich
Weizmann Institute/MPI Bonn
Analytic geometry over the field of one element
Elmar
Große-Klönne
HU Berlin
On the crystalline monodromy-weight conjecture for p-adically uniformized varieties
Anna
von Pippich
HU Berlin
Elliptic Eisenstein series for PSL_2(Z)
April 2008
Jürg
Kramer
HU Berlin
Spectral expansion of hyperbolic Eisenstein series
Jorge A.
Pineiro
City Univ. of New York
Heights and algebraic dynamics
February 2008
Emmanuel
Ullmo
Universite Paris-Sud, Orsay
Rational points on Shimura varieties
Jose Ignacio
Burgos Gil
Univ. of Barcelona
The arithmetic Riemann Roch theorem for closed immersions
January 2008
Sascha
Orlik
Univ. Leipzig
Äquivariante Vektorbündel auf Drinfelds Halbraum
Giancarlo
Urzua
Univ. of Michigan
Arrangements of curves and algebraic surfaces
Abstract.
We show a close relation between Chern and log Chern numbers of complex algebraic surfaces. In few words, given a log surface (Y,D) of a certain type, we prove that there exist smooth projective surfaces X with Chern ratio arbitrarily close to the log Chern ratio of (Y,D). The method is a "random" p-th root cover which exploits a large scale behavior of Dedekind sums and negative-regular continued fractions. We emphasize that the "random" hypothesis is necessary for this limit result. For certain divisors D, this construction controls the irregularity and/or the topological fundamental group of the new surfaces X. For example, we show how to obtain simply connected smooth projective surfaces, which come from the dual Hesse arrangement, with Chern ratio arbitrarily close to 8/3. In addition, by means of the Hirzebruch inequality for complex line arrangements, we show that this is the (unique) best result (closer to the Miyaoka-Yau bound 3) for complex line arrangements.
Amir
Dzambic
Univ. Frankfurt a. Main
Geometrie und Arithmetik falscher projektiver Ebenen
Jay
Jorgenson
CCNY
Zeta functions and determinants on discrete tori
Abstract.
For any integer m, we let mZ\Z denote a discrete circle, and we define a discrete torus to be a product of a finite number of discrete circles. Associated to the combinatorial Laplacian, one has a finite set of eigenvalues from which one can form the determinant, namely the product of the eigenvalues. We prove, under reasonably general assumptions, an asymptotic expansion of the determinant as the parameters m in the discrete circles tend to infinity. Specifically, we establish a "lead term" which solely involves the parameters of the discrete circles, and a "second order term" which is a modular form associated to the Kronecker limit problem for Epstein zeta functions. We show that the modular form is that which is obtained when defining the regularized determinant of the Laplacian on real tori, thus establishing a new connection between determinants and zeta regularized determinants, as well as discrete and real tori.
December 2007
Maria
Petkova
HU Berlin
Ball quotients curves with application in the coding theory
November 2007
Walter
Gubler
Univ. Dortmund, z.Z. HU Berlin
Tropische analytische Geometrie und die Bogomolov-Vermutung
Aleksander
Momot
ETH Zürich
Toroidally compactified ball quotient surfaces in small Kodaira dimension
Abstract.
As well as Hilbert modular surfaces, compact and compactified quotients $X = ar{\Gamma \\setminus \\mathbf{B}}$ of the open complex unit-ball $\\mathbf{B} \\subset \\mathbb{C}^2$ serve as the two-dimensional analog of modular curves. They are thus intimately connected with modular problems, but also provide interesting and extremal examples in the geography of surfaces. While Holzapfel has classified such surfaces for the case that they are defined by Picard modular ball-lattices $\\Gamma = \\mathbf{PU}(2,1; \\mathfrak{a})$, $\\mathfrak{a}$ an order in an imaginary quadratic number field, I aim a classification avoiding arithmetic conditions on Gamma. In the main part of the talk I will sketch the classification for kod(X)<= 0, q(X)> 0.
Gabor
Wiese
Univ. Duisburg-Essen, Campus Essen
Zur Erzeugung von Koeffizientenkörpern von Neuformen durch einen einzigen Hecke-Eigenwert
October 2007
Christian
Christensen
Univ. Dortmund, z.Z. HU Berlin
Der relative Satz von Schanuel