An Activity of the European Network Arithmetic Algebraic Geometry
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ABSTRACTS

• Joseph Ayoub
Homotopy t-structures on the categories of motives

I'll recal the Voevodsky's homotopy t-structure on the category of motives over a field and give two equivalent descriptions of its heart due to Deglise. Then I describe a generalisation of this t-structure for families of motives.
• Laurent Berger

Given a p-adic representation V, Fontaine has defined what it means for V to be crystalline, semistable, de Rham, etc. Now let X be a space parameterizing p-adic galois representations V_x. I will explain some of the things we can say about the subspaces corresponding to those x's for which V is crystalline, semistable, de Rham, etc. This is joint work with Pierre Colmez.
• José Ignacio Burgos Gil
On singular Bott-Chern forms

The singular Bott-Chern forms measure the failure of an exact Riemann-Roch theorem at the level of currents. They are the key ingredient in the definition of direct images of hermitian vector bundles under closed immersions and in the proof of the arithmetic Riemann-Roch theorem in Arakelov geometry for closed immersions. There are two definitions of singular Bott-Chern forms. The first due to Bismut, Gil et and Soulé uses the formalism of super connections. The second, due to Zha, is an adaptation of the original definition of Bott-Chern classes by Bott and Chern. In this talk we wil give an axiomatic characterization of singular Bott-Chern, which is similar to the characterization of Bott-Chern forms, but that depends on the choice of an arbitrary characteristic class. This characterization al ow us to give a new definition of singular Bott-Chern forms by means of the deformation to the normal cone technique and to compare the previous definitions of singular Bott-Chern forms.
• Daniel Caro
Splitting of F-complexes of arithmetic D-modules in overconvergent F-isocrystals
In order to construct a p-adic cohomology over schemes of characteristic p stable by Grothendieck's six operations, Berthelot has constructed arithmetic D-modules. In this talk, after recal ing its background, we wil show how overconvergent F-isocrystals can be described with the language of arithmetic D-modules. This enables us to define and check the fol owing property: for every overholonomic F-complex E of arithmetic D-modules over a variety X of characteristic p, there exists a splitting of X in local y closed subvarieties Xi such that the restrictions of E on Xi become much simpler, i.e., their cohomological spaces are associated with overconvergent F-isocrystals. This corresponds to an analogue of the splitting of l-adic constructible sheaves on X in smooth sheaves. We deduce straight away from this splitting a cohomological formula for L-functions associated with arithmetic D-modules and a p-adic 'Weil II'.
• Pierre-Henri Chaudouard
On the homology of truncated affine Springer fibers
The fundamental lemma of Langlands-Shelstad is the cornerstone of the theory of endoscopy. For unitary groups, Laumon and Ngô have proved it by geometric methods. In general, it remains a deep conjecture. In his work on the stabilization of the trace formula, Arthur has introduced a generalization, the weighted fundamental lemma. This aim of this talk is to give it a geometric interpretation. This is joint work with Gérard Laumon.
• Marianna Fornasiero
Comparison Theorems in Logarithmic Cohomology

Given a variety defined over the complex numbers (even singular), the problem of defining a suitable De Rham cohomology and its comparison with other cohomologies was posed several years ago by many authors (see for example Grothendieck, Hartshorne, Deligne...). This has led (among other reasons) to the definition of a infinitesimal sites and finally to the comparison between the Algebraic De Rham Cohomology of a singular scheme, its Infinitesimal (or Crystalline) Cohomology, and the Betti Cohomology of its associated analytic space (see works of Grothendieck, Hartshorne, Deligne, Herrera-Lieberman, Illusie, Berthelot, Ogus, and others). In this talk, we propose a generalization of these results and definitions to the logarithmic setting (characteristic zero, over the complex numbers), inspired by works of K. Kato, C. Nakayama, L. Illusie, A. Ogus, A. Shiho. Namely, given S=Spec(C) (endowed with the trivial log structure) and an fs not necessary (ideally) log smooth log scheme Y over S, we analyze the connection between the Log De Rham Cohomology of Y, its Log Infinitesimal Cohomology H^{^.}(Y^{log}_{inf}, O_{Y^{log}_{inf}}), and its Log Betti Cohomology, which is defined as the Cohomology of its associated Kato-Nakayama topological space Y^{an}_{log}. We first show that they are isomorphic under the hypothesis that there exists an exact closed immersion of Y into a log smooth log scheme X over S, then we extend these comparison theorems to any log scheme over S, by working with good embedding systems, log formal tubes and by using descent properties. These results generalize the (ideally) log smooth log case developed by Kato-Nakayama. (Adjoint work with B. Chiarellotto).
• Michael Harris
Automorphic Galois representations and the Sato-Tate Conjecture

I will report on my recent proof with Taylor, Clozel, and Shepherd-Barron of the Sato-Tate Conjecture for elliptic curves over Q with non-integral j-invariant. The theorem is a consequence of the proof of potential automorphy of even-dimensional symmetric powers of the Tate module of such an elliptic curve. This work will be presented in the context of the conjectured Langlands correspondence between Galois representations and automorphic representations.
• Marco Hien
Periods for irregular singular connections on surfaces

Generalizing a definition of S. Bloch and H. Esnault for the case of curves, we define homology groups for irregular singular connections on complex surfaces together with a period pairing between these and the algebraic de Rham cohomology of the connection. We prove the non-degeneracy of the pairing and comment on new phenomena appearing in the two-dimensional situation.
• Jan Kohlhaase
Cohomology of locally analytic representations

We define a cohomology theory for locally analytic representations of p-adic Lie groups in the sense of Schneider/Teitelbaum. This theory allows us to prove versions of Shapiro's lemma and Hochschild-Serre's spectral sequence, naturally leading to a generalized notion of supercuspidality. As an application we indicate how to compute the group of extensions between two locally analytic principal series representations.
• Matteo Longo
Iwasawa's Main Conjecture for Hilbert Modular Forms over anticyclotomic Z_p-extensions.

I will discuss the following result on anticyclotomic Iwasawa's main conjecture for Hilbert modular forms: Let f be a Hilbert modular form over a totally real field F and p a rational prime. Under suitable assumptions, the characteristic power series C_p(f,K_\infty) of the Pontryagin dual of the Selmer group attached to f over certain Z_p-anticyclotomic extension K_\infty/F divides the anticyclotomic p-adic L-function L_p(f,K_\infty). Here C_p(f,K_\infty) and L_p(f,K_\infty) are elements of the Iwasawa algebra of Gal(K_\infty/K)\simeq Z_p^d, where K/F is a suitable quadratic totally imaginary extension and d is an integer. I will also discuss some applications to modular abelian varieties of GL_2-type and some relations between p-adic L-functions associated to different anticyclotomic extensions K_\infty and K'_\infty. This is a generalization by the previous work by Bertolini and Darmon for elliptic modular forms.
• Philippe Michel
Equidistribution of periodic torus orbits on spaces of lattices

We explain several recent equidistribution results for periodic orbits of the diagonal torus on the space of lattices. This is joint work with M. Einsiedler, E. Lindenstrauss and A. Venkatesh
• Fabrice Orgogozo
p-dimension of henselian fields: an application of Ofer Gabber's algebraization technic.

If A is an henselian excellent integral ring of characteristic (0,p), whose fraction field is K and residue field is k, it has been conjectured by Kazuya Katô, who also proved it for discrete valuation rings, that the p-cohomological dimension of K is (almost) equal to dim(A) plus the p-rank of k. (It could be this number plus one.) We will explain the proof of the above formula and its analogue in equal characteristic. The main ingredients are a result of algebraization due to Ofer Gabber (and recently used by him in his proof of the finiteness of direct images for the étale topology), as well as K. Katô's formula.
• Kartik Prasanna
Elliptic curves, quadratic twists and p-(in)divisibility of L-values

Let E be an elliptic curve over the rationals and L(E,s) the L-function associated to E. A famous theorem of Waldspurger states that there exists a quadratic discriminant d such that the central value L(E_d, 1) is nonzero, where E_d is the d'th quadratic twist of E. This theorem along with its generalizations has many applications, most notably to the Birch and Swinnerton-Dyer conjecture in the case of rank 1 elliptic curves. I will explain a conjectural "mod p" version of Waldspurger's theorem and some related results that are obtained by studying the p-adic properties of the Shimura correspondence. As a corollary, one also obtains interesting information about p-indivisibility of orders of Tate-Shafarevich groups of quadratic twists.
• Takeshi Saito
Ramification of schemes over a local field

For l-adic etale sheaves on schemes over a local field, we define the Swan class in the dimension 0 part of the K-group of coherent sheaves supported on the boundary, as an invariant measuring wild ramification. It satisfies a Riemann-Roch formula for higher direct image. It is an analogue of the Grothendieck-Ogg-Shafarevich formula and is a generalization of the conductor formula of Bloch.
• Yiannis Sakellaridis
On the unramified spectrum of spherical varieties over p-adic fields

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the "Langlands dual" group. We generalize this description to an arbitrary spherical variety X of G as fol ows: Irreducible quotients of the "unramified" Bernstein component of C_c^\infty (X) are in natural almost bijection with (a number of copies of) the quotient of a complex torus by the "little Weyl group'" of X. This leads to a weak analog of results of D. Gaitsgory and D. Nadler on the Hecke module of unramified vectors, and an understanding of the phenomenon that representations "distinguished" by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given to the action, defined by F. Knop, of the Weyl group on the set of Borel orbits."
• Emmanuel Ullmo
Around the André-Oort conjecture

We'll explain recent progress on the André-Oort conjecture including Galois and ergodic techniques.
• Yakov Varshavsky
Moduli of F-bundles and global Langlands correspondence over function fields for split reductive groups.

Langlands conjectured that there is a correspondence between automorphic representations of G and homomorphisms from the Galois group to the Langlands' dual group of G. In my talk I wil recal this conjecture, describe an approach to its proof, and formulate a partial result (joint with David Kazhdan).
• Eva Viehmann
Connected components of affine Deligne-Lusztig varieties

Affine Deligne-Lusztig varieties are analogues of the classical Deligne-Lusztig varieties inside the affine flag manifold or the affine Grassmannian. They are also related to the reduction modulo p of Rapoport-Zink spaces. This talk is on recent results on the global structure of affine Deligne-Lusztig varieties inside the affine Grassmannian associated to a split reductive group, and primarily on their sets of connected components.
• Annette Werner
Compactifications of Bruhat-Tits buildings and linear representations

Let G be a semi-simple group over a non-archimedean local field. For every geometrical y irreducible linear representation of G we construct a compactification of the Bruhat-Tits building X(G) associated to G. This gives a finite family of compactifications of X(G) containing the polyhedral one. Our compactifications can be seen as non-archimedean analogs of the different Satake compactifications of symmetric spaces.
• Jean-Pierre Wintenberger
On Serre's conjecture

I will give an outline of a proof of Serre's modularity conjecture for an irreducible odd representation of the Galois group of Q in the linear group of dimension 2 over a finite field F. If the characteristic of F is 2, or if the representation is ramified at 2, the proof is conditional to a lifting modularity theorem for 2-adic Galois representations.

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