# Arithmetic, Motives and Moduli Spaces

Arithmetic, Motives and Moduli Spaces est un colloque organisé à l'IHP du 9 au 13 janvier 2012 par Olivier Debarre (ENS), Hélène Esnault (Essen), Claire Voisin (IMJ) et Olivier Wittenberg (ENS) à l'occasion du séjour à la Fondation d'Hélène Esnault, lauréate de la Chaire d'Excellence 2011.

### Les conférences

Holoedry of Euclidean lattices and quasi-homogeneous varieties, par Yves André (ENS)

**Résumé** : We explain how symmetries of a Euclidean lattice and its tensor powers can be controlled in terms of a reductive group and an affine quasi-homogeneous variety canonically attached to the lattice (construction inspired by the theory of motives).

Holoedry of Euclidean lattices and... *par Sciences_Maths_Paris*

De Rham fundamental group of a punctured elliptic curv, par Francis Brown (Jussieu)

Vidéo non disponible.

Cohomology classes on moduli of abelian varieties, par Samuel Grushevsky (Stony Brook)

**Résumé** : We compute the Chow and cohomology classes of some geometrically defined loci in the moduli space of principally polarized abelian varieties. These computations could be viewed as a step towards defining an extended tautological ring: the subring of cohomology of a suitable toroidal compactification of the moduli space containing the Chern classes of the Hodge bundle and some boundary classes.

Cohomology classes on moduli of abelian... *par Sciences_Maths_Paris*

Unlikely Intersections in Products of Modular Curves and Abelian Schemes, par Philipp Habegger (Frankfurt)

**Résumé :** A general conjecture of Pink predicts how a fixed subvariety of a mixed Shimura variety meets special subvarieties of unusually large codimension. A closely related conjecture was formulated earlier by Zilber for semi-abelian varieties. I will present results into the direction of this conjecture in the particular setting of abelian schemes and a product of modular curves. Some of this is joint work with Jonathan Pila.

Unlikely Intersections in Products of Modular... *par Sciences_Maths_Paris*

On motivic classes of moduli spaces of Higgs bundles**,** par Jochen Heinloth (Essen)

**Résumé** : Motivated by a series of conjectures on the cohomology of the moduli space of Higgs bundles on curves we found a cut-and-paste approach to the computation of the cohomology of these spaces. This is joint work with O. Garcia- Prada and A. Schmitt.

On motivic classes of moduli spaces of Higgs... *par Sciences_Maths_Paris*

p-adic periods for reductive groups, par Annette Huber (Freiburg)

**Résumé** : (joint work with G. Kings, Regensburg) We propose a definition of p-adic periods for reductive groups that replaces the complex comparison isomorphism between de Rham cohomology and singular cohomology by the Lazard isomorphism between Lie group cohomology and continuous groups cohomology. This is in analogy to Borel's approach at the infinite fiber. We were able to show that these p-adic periods are the correct factors turning Borel's formula for values of Dedekind zeta function into an integral formula. We can also show that these p-adic periods compute the Soulé regulator, ie., the étale Chern class that occurs in the Bloch- Kato conjecture.

Vidéo non disponible.

p-adic deformation of cycle classes, par Moritz Kerz (Regensburg)

**Résumé** : We study the deformation properties of cycle classes for a p-adic smooth projective scheme over the ring of Witt vectors. The expected deformation is described by the p-adic variational Hodge conjecture. (Joint work with S. Bloch and H. Esnault.)

p-adic deformation of cycle classes, par Moritz... *par Sciences_Maths_Paris*

Symmetric differentials and fundamental group, par Bruno Klingler (Jussieu)

**Résumé** : Classical Hodge theory says that the sheaves of exterior differentials control the usual Betti cohomology of a smooth complex projective variety X. Is there a similar relation between the sheaves of symmetric differentials and the topology of X ? In this talk I will explain how symmetric differentials control finite dimensional representations of the topological fundamental group of X. Using automorphic forms this leads to new striking rigidity properties for fundamental groups of certain ball quotients.

Symmetric differentials and fundamental group... *par Sciences_Maths_Paris*

On a positive equicharacteristic version of the p-curvature conjecture, par Adrian Langer (Warsaw/Essen)

**Résumé** : I would like to talk about the p-curvature conjecture. The equicharacteristic zero version of this conjecture was obtained by Yves Andre. I will focus on possible generalizations to positive equicharacteristic. This is a report on a joint work with Hélène Esnault.

On a positive equicharacteristic version of the... *par Sciences_Maths_Paris*

Semi-algebraic horizontal subvarieties of Calabi-Yau type, par Radu Laza (Stony Brook)

**Résumé** : Except the classical cases of abelian varieties and K3 surfaces, the images of period maps for families of algebraic varieties satisfy non-trivial Griffiths' transversality relations. It is of interest to understand these images of period maps, especially for Calabi-Yau threefolds. In this talk, we will discuss the case when the images of period maps can be described algebraically. Specifically, we show that if a horizontal subvariety Z of a period domain D is semi-algebraic and is stabilized by a large discrete group, then Z is a Hermitian symmetric domain with a totally geodesic embedding into the period domain D. Additionally we classify all the semi-algebraic cases for variations of Hodge structures of Calabi-Yau type (complementing some earlier work of Gross and Sheng-Zuo). This is joint work with R. Friedman.

Semi-algebraic horizontal subvarieties of... *par Sciences_Maths_Paris*

Finiteness of K3 surfaces and the Tate conjecture, par Davesh Maulik (MIT)

**Résumé** : Given a class of varieties, it is natural to ask if it has only finitely many members defined over each finite field. I will discuss this question for K3 surfaces and explain the following result: one has such a finiteness statement for K3 surfaces if and only if the Tate conjecture holds for K3 surfaces. This is joint work with Max Lieblich and Andrew Snowden.

Finiteness of K3 surfaces and the Tate... *par Sciences_Maths_Paris*

Construction of Kobayashi curves, par Martin Möller (Frankfurt)

**Résumé :** Kobayashi curves are algebraic curves that are totally geodesic for the Kobayashi metric. On Hilbert modular surfaces these curves are rigid and thus interesting from a geometric and arithmetic perspective. Kobayashi curves include the well-known Hirzebruch-Zagier cycles but besides that there are very few techniques to construct them.

We present a new method to exhibit Kobayashi curves using derivatives of theta functions. This solves also the problem of determining the Euler characteristic of some Teichmueller curves constructed by McMullen.

Construction of Kobayashi curves, by Martin Möller *par Sciences_Maths_Paris*

Motivic decomposition of abelian schemes, par Ben Moonen (Amsterdam)

**Résumé** : I will first review some classical facts, due to Beauville, Deninger, Murre and Kuennemann, related to the decomposition of the Chow motive of an abelian scheme.

After that I will show, using some basic representation theory, that there is a natural refinement of these decompositions for abelian schemes that have non-trivial endomorphisms.

Motivic decomposition of abelian schemes, by Ben... *par Sciences_Maths_Paris*

Introduction to periods and Nori motives, par Stefan Müller-Stach (Mainz)

**Résumé** : This is a report on recent joint work with Annette Huber on the relation between the period torsor and the Tannaka group of Nori mixed motives, following some ideas of Kontsevich and Nori Vidéo non disponible.

Néron component groups of wildly ramified abelian varieties, par Johannes Nicaise (Leuven)

**Résumé** : I will present some results on the behaviour of Néron component groups of abelian varieties under ramified base change, with special emphasis on wildly ramified abelian varieties. The main idea is that Néron models behave well under base change if the extension is "sufficiently orthogonal" to the minimal extension where the abelian variety acquires semi-abelian reduction. The hardest case is the case of abelian varieties with potential good reduction; I will discuss some open problems and possible approaches in that setting.

Néron component groups of wildly ramified... *par Sciences_Maths_Paris*

A positive characteristic analog of Manin's theorem of the kernel, par Damian Rössler (Toulouse)

**Résumé** : Let A be an abelian variety defined over the function field K of a smooth curve defined over a finite field of characteristic p>0. We shall present some results and conjectures on the structure of the set $cap_{kgeq 0}p^kcdot A(K^{rm sep})$. In particular, we shall show that it consists of torsion points, as conjectured by A. Pillay.

This can be viewed as an analog of Manin's theorem of the kernel.

A positive characteristic analog of Manin's... *par Sciences_Maths_Paris*

Higher nonabelian cohomology of algebraic varieties, par Carlos Simpson (Nice)

**Résumé** : The aim of this talk is to give an introduction to nonabelian cohomology of complex algebraic varieties via higher stacks, with an emphasis on the possibility of doing Hodge theory and a discussion of where some of the problems currently lie.

Higher nonabelian cohomology of algebraic... *par Sciences_Maths_Paris*

Rational points on algebraic varieties over function fields , par Yuri Tschinkel (NYU)

**Résumé** : I will discuss various approaches to the construction of sections of fibrations over curves and higher-dimensional bases.

Vidéo non disponible.

Buildings and non-archimedean flag varieties, par Annette Werner (Frankfurt)

**Résumé** : In joint work with Bertrand Rémy and Amaury Thuillier it was shown that Bruhat-Tits buildings can be embedded in Berkovich analytic flag varieties. The image of the building is contained in an open subset which in the case of projective space is Drinfeld's well-known p-adic upper half plane. We will discuss these embeddings and investigate degenerations of flag varieties defined with subcomplexes of the building.

Buildings and non-archimedean flag varieties, by... *par Sciences_Maths_Paris*

A hyperbolic Ax-Lindemann theorem in the cocompact case, par Andrei Yafaev (UCL)

**Résumé*** :* This is a joint work with Emmanuel Ullmo.This work is motivated by J.Pila's strategy to prove the André-Oort conjecture. One ingredient in the strategy is the following conjecture: Let S be a Shimura variety uniformised by a symmetric space X. Let V be an algebraic subvariety of S. Maximal algebraic subvarieties of the preimage of V in X are precisely the components of the preimages of weakly special subvarieties contained in V. We will explain the proof of this conjecture in the case where S is compact.

A hyperbolic Ax-Lindemann theorem in the... *par Sciences_Maths_Paris*