An Activity of the European Network Arithmetic Algebraic Geometry



ABSTRACTS Homotopy tstructures on the categories of motives I'll recal the Voevodsky's homotopy tstructure 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 tstructure for families of motives. Families of padic galois representations Given a padic representation V, Fontaine has defined what it means for V to be crystalline, semistable, de Rham, etc. Now let X be a space parameterizing padic 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. On singular BottChern forms The singular BottChern forms measure the failure of an exact RiemannRoch 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 RiemannRoch theorem in Arakelov geometry for closed immersions. There are two definitions of singular BottChern 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 BottChern classes by Bott and Chern. In this talk we wil give an axiomatic characterization of singular BottChern, which is similar to the characterization of BottChern forms, but that depends on the choice of an arbitrary characteristic class. This characterization al ow us to give a new definition of singular BottChern forms by means of the deformation to the normal cone technique and to compare the previous definitions of singular BottChern forms. Splitting of Fcomplexes of arithmetic Dmodules in overconvergent Fisocrystals In order to construct a padic cohomology over schemes of characteristic p stable by Grothendieck's six operations, Berthelot has constructed arithmetic Dmodules. In this talk, after recal ing its background, we wil show how overconvergent Fisocrystals can be described with the language of arithmetic Dmodules. This enables us to define and check the fol owing property: for every overholonomic Fcomplex E of arithmetic Dmodules 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 Fisocrystals. This corresponds to an analogue of the splitting of ladic constructible sheaves on X in smooth sheaves. We deduce straight away from this splitting a cohomological formula for Lfunctions associated with arithmetic Dmodules and a padic 'Weil II'. On the homology of truncated affine Springer fibers The fundamental lemma of LanglandsShelstad 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. 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, HerreraLieberman, 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 KatoNakayama 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 KatoNakayama. (Adjoint work with B. Chiarellotto). Automorphic Galois representations and the SatoTate Conjecture I will report on my recent proof with Taylor, Clozel, and ShepherdBarron of the SatoTate Conjecture for elliptic curves over Q with nonintegral jinvariant. The theorem is a consequence of the proof of potential automorphy of evendimensional 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. 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 nondegeneracy of the pairing and comment on new phenomena appearing in the twodimensional situation. Cohomology of locally analytic representations We define a cohomology theory for locally analytic representations of padic Lie groups in the sense of Schneider/Teitelbaum. This theory allows us to prove versions of Shapiro's lemma and HochschildSerre'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. Iwasawa's Main Conjecture for Hilbert Modular Forms over anticyclotomic Z_pextensions. 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_panticyclotomic extension K_\infty/F divides the anticyclotomic padic Lfunction 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_2type and some relations between padic Lfunctions 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. 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 pdimension 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 pcohomological dimension of K is (almost) equal to dim(A) plus the prank 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. Elliptic curves, quadratic twists and p(in)divisibility of Lvalues Let E be an elliptic curve over the rationals and L(E,s) the Lfunction 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 SwinnertonDyer 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 padic properties of the Shimura correspondence. As a corollary, one also obtains interesting information about pindivisibility of orders of TateShafarevich groups of quadratic twists. Ramification of schemes over a local field For ladic etale sheaves on schemes over a local field, we define the Swan class in the dimension 0 part of the Kgroup of coherent sheaves supported on the boundary, as an invariant measuring wild ramification. It satisfies a RiemannRoch formula for higher direct image. It is an analogue of the GrothendieckOggShafarevich formula and is a generalization of the conductor formula of Bloch. On the unramified spectrum of spherical varieties over padic 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 padic reductive group G over a local nonarchimedean 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." Around the AndréOort conjecture We'll explain recent progress on the AndréOort conjecture including Galois and ergodic techniques. Moduli of Fbundles 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). Connected components of affine DeligneLusztig varieties Affine DeligneLusztig varieties are analogues of the classical DeligneLusztig varieties inside the affine flag manifold or the affine Grassmannian. They are also related to the reduction modulo p of RapoportZink spaces. This talk is on recent results on the global structure of affine DeligneLusztig varieties inside the affine Grassmannian associated to a split reductive group, and primarily on their sets of connected components. Compactifications of BruhatTits buildings and linear representations Let G be a semisimple group over a nonarchimedean local field. For every geometrical y irreducible linear representation of G we construct a compactification of the BruhatTits 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 nonarchimedean analogs of the different Satake compactifications of symmetric spaces. 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 2adic Galois representations.

