No. SM-187 24/06/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Higher Ramanujan equations and periods of abelian varieties
Tiago FONSECA, Université Paris-Sud
Abstract: The Ramanujan equations are some algebraic differential equations satisfied by the classical Eisenstein series E_2, E_4, E_6. These equations play a pivotal role in the proof of Nesterenko's celebrated theorem on the algebraic independence of values of Eisenstein series, which gives in particular a lower bound on the transcendence degree of fields of periods of elliptic curves. Motivated by the problem of extending the methods of Nesterenko to other settings, we shall explain in this talk how to generalize Ramanujan's equations to higher dimensions via a geometric approach, and how the values of a particular solution of these equations relate with periods of abelian varieties.
No. SM-186 24/06/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Representation theory of p-adic groups and type theory
Peiyi CUI (崔沛仪) , Université de Rennes 1
Abstract: Given a reductive p-adic group G(F), according to parabolic induction, it is very natural to divide its irreducible representations by supercuspidality. However, in 1977, Howe gave a totally different opinion: He introduced the notion of “types” and suggested that considering the restriction to some open compact subgroups should be a more essential method when study complex smooth representations. This talk will start with the background and some convincible results, later we will see the description of a supercuspidal representation in the language of type, at the end I will show some further use in modular representation theory.
No. SM-185 17/06/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Entropy of an autoequivalence on Calabi-Yau manifolds
Yu-Wei FAN, Harvard University
Abstract: In the first part of the talk, we will discuss homological mirror symmetry for elliptic curves, which leads to an addition formula for theta functions as a byproduct. In the second part, we will define the categorical entropy of an autoequivalence and disprove a conjecture by Kikuta and Takahashi. Finally, we will discuss how these two parts are related.
No. SM-184 03/06/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Autour d'une conjecture de Kato et Kuzumaki
Diego IZQUIERDO, Université Paris-Sud
Résumé: Dans cet exposé je vais parler d'une conjecture de Kato et Kuzumaki dont le but est de donner une caractérisation de la dimension cohomologique d'un corps en termes diophantiens, en faisant interagir les hypersurfaces projectives de petit degré et la K-théorie de Milnor. Si la conjecture est fausse en général, elle reste totalement ouverte pour les corps qui apparaissent usuellement en géométrie algébrique ou en théorie des nombres. Je présenterai les différents résultats qui sont connus actuellement et en démontrerai certains.
No. SM-183 03/06/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Sum-product phenomenon in representations of Lie groups
Weikun HE (何伟鲲), Université Paris-Sud
Abstract: In rings we can observe interesting interactions between addition and multiplication. Typically if a subset grows slowly under addition then it grows fast under multiplication and vice versa, unless it is close to a subring. Multiplicative subgroups behave like random subsets viewed in the additive group. These incompatibilities between additive and multiplicative structures are called sum-product phenomenon. We will briefly introduce classical sum-product results and then present a similar phenomenon in representations of Lie groups (joint work with Nicolas de Saxcé).
No. SM-182 27/05/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
L'étude des réseaux euclidiens appliquée à la compréhension des formes automorphes des groupes linéaires
Thomas MEGARBANE, Université Paris-Sud
Résumé: Dans le but de mieux comprendre les représentations automorphes algébriques des groupes linéaires récemment découvertes par Chenevier et Renard, on cherche à obtenir des informations sur leurs paramètres de Satake. Notre point de départ est la théorie d'Arthur, qui permet de voir ces paramètres grâce aux paramètres de Satake de représentations automorphes discrètes des groupes spéciaux orthogonaux de réseaux bien choisis. On étudie dans un premier temps les opérateurs de Hecke associés aux voisins de Knesser de certains réseaux : suivant Gross, on arrive ainsi à calculer la trace des premiers paramètres de Satake des représentations des groupes linéaires qui nous intéressent. Dans un second temps, on peut étudier ces mêmes opérateurs agissant sur les classes d'isomorphisme de réseaux, ce que l'on fait en dimension 23 et 25. On en déduit des congruences, du type "congruence de Ramanujan" ou "congruence de Harder", pour les traces des paramètres de Satake des représentations des groupes linéaires considérées.
No. SM-181 27/05/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Equidistribution of elliptic fibrations in families of K3 surfaces
Salim TAYOU, Université Paris-Sud
Abstract: In this talk, I will explain a result on equidistribution of elliptic fibrations in proper families of polarized K3 surfaces. Under SYZ conjecture, the similar equidistribution result also holds for proper families of polarized Hyperkähler manifolds. The proof is based on Borcherds' construction of an automorphic form on a complex Shimura variety associated to a lattice of signature (2,n) and some results from dynamics on homogeneous spaces proved by Eskin-Oh.
No. SM-180 06/05/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Images of toric varieties and liftability of the Frobenius morphism
Piotr ACHINGER, IHES
Abstract: The celebrated proof of the Hartshorne conjecture byShigefumi Mori allowed for the study of the geometry of higher dimensional varieties through the analysis of deformations of rational curves. One of the many applications of Mori's results was Lazarsfeld's positive answer to the conjecture of Remmert and Van de Ven which states that the only smooth variety that the projective space can map surjectively onto is the projective space itself. Motivated by this result, a similar problem has been considered for other kinds of varieties such as abelian varieties (Demailly-Hwang-Mok-Peternell) or toric varieties (Occhetta-Wiśniewski). In my talk, I would like to present a completely new perspective on the problem coming from the study of Frobenius lifts in positive characteristic. This is based on a joint project with Jakub Witaszek and Maciej Zdanowicz.
No. SM-179 06/05/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Torsion pour les variétés abéliennes de type III
Victoria Cantoral FARFAN, IMJ-PRG
Résumé: 179_Victoria CANTORAL-FARFAN_Resume
No. SM-178 29/04/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
On special values of L-functions and the Deligne conjecture
Jie LIN (林洁), IHES
Abstract: The aim of this talk is to introduce the Deligne conjecture on special values of L-functions. We will start from Dirichlet L-functions, and then define automorphic L-functions. We will talk about the Deligne conjecture and it's automorphic variant after.
No. SM-177 29/04/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Groupe de Brauer invariant et obstruction de descente itérée
Yang CAO (曹阳), Université Paris 11
Résumé: Pour une variété quasi-projective, lisse, géométriquement intègre sur un corps de nombre k, on montre que l'obstruction de descente itérée est équivalente à l'obstruction de descente. Ceci répond à une question ouverte de Poonen. L'idée clé est la notion de sous-groupe de Brauer invariant et la notion d'obstruction de Brauer-Manin invariante étale pour une k-variété munie d'une action d'un groupe linéaire connexe.
No. SM-176 01/04/2017 16:30 ~ 18:00 Amphithéâtre Léon Motchane, IHES
Overconvergence of etale (\varphi, \tau)-modules
Hui GAO (高辉), University of Helsinki
Abstract: The category of etale (\varphi, \tau)-modules, similar as the category of etale (\varphi, \Gamma)-modules, is equivalent to the category of p-adic Galois representations. A classical theorem of Cherbonnier-Colmez says that all etale (\varphi, \Gamma)-modules are overconvergent. In this talk, we show that all etale (\varphi, \tau)-modules are also overconvergent. Our method is completely different from that of Cherbonnier-Colmez. The key idea is a certain crystalline approximation technique. This is joint work with Tong Liu.
No. SM-175 01/04/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Non abelian Hodge theory in positive characteristic (after Ogus-Vologodsky)
Daxin XU (许大昕), IHES
Abstract: In their seminal work, Ogus and Vologodsky developed an analogue of Simpson's correspondence for modules with integrable connection in positive characteristic. They also extended Deligne-Illusie's result for de Rham cohomology with coefficients.
In this talk, I will introduce this theory and its application to studying relative Fontaine-Laffaille modules and semi-stable Higgs bundles.
No. SM-174 18/03/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
On the counting problems of l-adic local systems and vector bundles over a curve
Hongjie YU (余红杰), Université Paris 7
Abstract: Arthur’s non-invariant trace formula is an intermediate product of his final trace formula. It’s complicated but has the advantage of being explicit. It can be showed that an analogue of Arthur’s truncated trace in the Lie algebra setting is directly related to counting problems of Hitchin’s bundles. For Arthur’s original non-invariant trace formula, the geometric side will have similar explanations, while spectral side is in connection with local systems on a curve. In this talk, I will show some results on the counting problems of l-adic local systems and vector bundles over a curve over a finite field using Arthur’s non-invariant trace formula.
No. SM-173 04/03/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Images of Galois representations associated to Hida families
Jaclyn LANG, Université Paris 13
Abstract: We explain a sense in which Galois representations associated to non-CM Hida families have large images. This is analogous to results of Ribet and Momose for Galois representations associated to classical modular forms. In particular, we show how extra twists of the Hida family decreases the size of the image.
No. SM-172 04/03/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
A survey of the Bernstein decomposition of mod l representations of p-adic general linear groups (l different from p)
Peiyi CUI (崔沛仪) , Université de Rennes 1
Abstract: The first part of my talk will consider representations of finite general linear groups. By Deligne-Luztig theory, we have the classification of cuspidal representations with characteristics 0, and by the truth of Geck’s conjecture in this case, we can use reduction mod l to relate the cuspidal representations with char=0 to the cuspidal mod l representations, even to classify all the supercuspidal representations.
The second part will consider representations of p-adic general linear groups. Thanks to the works of Bushnell and Kutzko, we could use type theory (especially cuspidal types) to understand the (cuspidal) representations with char=0 or l, which is partly based on the acknowledge of representations of finite general linear groups. Then there will be a theorem to give a description of supercuspidal support of cuspidal mod l representations in the language of type theory, which is parallel to the results in the first part. Finally we can see the statement of Bernstein decomposition of mod l representations.
No. SM-171 18/02/2017 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Arithmetic families of (\varphi, \Gamma)-modules
Ildar GAISIN, IHES
Abstract: Let A be a Q_p-affinoid algebra in the sense of Tate. We propose a p-adic Langlands correspondence in families: For a <<quasi-regular>> trianguline (\varphi, \Gamma)-module of dimension 2 over the relative Robba ring R_A, we construct a locally analytic GL_2(Q_p)-representation in A-modules. This is joint work with Joaquin Rodrigues.
No. SM-170 18/02/2017 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Generalized Multiple Zeta Values on number fields
Xiaohua AI (艾小华) , Université Paris 6
Abstract: Multiple Dedekind zeta values (MDZV), which should be analogues of MZV over a number field K, have not been defined. Ivan Horozov gave an attempt of his definition of MDZV over a totally real field F by using a new type of iterated integrals. Inspired by Goncharov’s theory of Hodge correlator and the plectic principle due to Nekovar and Scholl, we propose another method to give a natural and systematic definition of generalized MZV over a totally real field F. We proved that the generalised MZV over a totally real field F can be written a linear combination of classical MZVs. In fact, our definition can be generalized to any number field, however, some effort should be added. For example, we can give generalized MZV over an imaginary quadratic field, we will give the Fourier expansion of our generalized MZV, which could be related to some non-holomorphic modular forms.
No. SM-169 10/12/2016 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
Counting Rational Points in Arithmetic Varieties
Chunhui LIU (刘春晖), Université Paris 7
Abstract: By the slope method in Arakelov geometry, we can construct a family of hypersurfaces which cover the rational points of bounded height on an arithmetic variety but don't contain the generic point of this variety. By estimating some invariants of Arakelov geometry, we can control the number and the maximal degree of this family of auxiliary hypersurfaces explicitly. In this talk, I will explain the method of studying the problem of counting rational points by the approach of Arakelov geometry.
In the above argument, it is important to consider an explicit upper bound of a counting function of multiplicity of rational points in a projective hypersurface over a finite field. I will give out such an upper bound to describe the complexity of singular locus of reduced projective hypersurfaces (Arxiv: 1606.09337).
No. SM-168 10/12/2016 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Jacobi sums and their applications to modulo p local Langlands correspondence
Zicheng QIAN (钱子诚), Université Paris-Sud
Abstract: We introduce the conjectural modulo p local Langlands correspondence. The first step towards this correspondence is the so called Serre weight conjecture. On the other hand, it is difficult to study the socle filtration of mod p principal series of GLn(Fp). We will introduce a family of group operators called Jacobi sums and explain how can we essentially reduce a relatively easy direction of modulo p local Langlands correspondence to the Serre weight conjecture, using Jacobi sums. We will explain GL3 as an example.
No. SM-167 26/11/2016 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
p-adic Integration and its Applications
Daniel KRIZ, Princeton university
Abstract: In this talk we will introduce Coleman's theory of integration in locally analytic functions on rigid analytic spaces and discuss some of its applications to the study of rational points on abelian varieties and other related questions. P-adic integration has been used in previous work of myself and recent work of Chao Li and myself to verify the Birch and Swinnerton-Dyer conjecture for large families of elliptic curves (and abelian varieties of GL_2 type) over Q. For example, we show that for any elliptic curve with rational 3-isogeny, a positive proportion of its quadratic twists have algebraic and analytic rank equal to 0 (resp. 1), thus verifying Goldfeld's conjecture for these curves. More generally, p-adic integration in certain cases allows one to prove the weak Beilinson-Bloch conjecture for Rankin-Selberg motives (and combined with previous work of Nekovar, even verifying the full conjecture under certain assumptions). Underlying the strategies of these proofs is the theory of p-adic L-functions, and if time permits I will also discuss some constructions of such functions.
No. SM-166 26/11/2016 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Espaces de Banach-Colmez
Arthur-César Le BRAS, ENS
Abstract: La catégorie des espaces de Banach-Colmez a été introduite par Colmez pour donner une nouvelle preuve de la conjecture « faiblement admissible implique admissible » de Fontaine. Dans la première moitié de l’exposé, j’expliquerai comment l’on peut interpréter cette catégorie en termes de la courbe de Fargues-Fontaine, que je présenterai. Dans la seconde, j’expliquerai une preuve récente, due à Fargues, de la théorie du corps de classes local par voie géométrique : les espaces de Banach-Colmez (généralisés) y jouent un rôle crucial.
No. SM-165 29/10/2016 16:30 ~ 18:30 Amphithéâtre Léon Motchane, IHES
A New Northcott Property for Faltings' Heights
Lucia MOCZ, Princeton University
Abstract: We develop explicit techniques using tools from integral $p$-adic Hodge theory to study the change in Faltings' height within an isogeny class of CM abelian varieties. Assuming the Colmez conjecture, this results in a new Northcott property for Faltings' heights for CM points. On the Hilbert modular variety we are moreover able to develop a Colmez-type formula for the Faltings' height of all CM points. In this talk we will focus on and emphasize the background from $p$-adic Hodge theory, namely Kisin's correspondence between integral crystalline representations and $\mathfrak{S}$-modules, and show how to use this correspondence to interpret the deformation theory of $p$-divisible groups to perform computations in the Arakelov setting.
No. SM-164 29/10/2016 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Cohomologie des fibrés en droites sur G/B
Linyuan LIU (刘琳媛), Université Paris 6
Résumé: Soit G un groupe algébrique réductif sur un corps algébriquement clos de caractère p>0. On fixe un tore maximal T et un sous-groupe de Borel B contenant T correspondant aux racines négatives. Pour chaque élément dans X(T) (le groupe des caractères de T), on peut définir une fibré en droites sur G/B. On s’intéresse aux groupes de cohomologie de cette fibré en droites. On commencera par étudier l’annulation de H^1.
No. SM-163 15/10/2016 16:30 ~ 18:39 Amphithéâtre Léon Motchane, IHES
p-adic Universal Covers of an Abelian Variety and the Manin-Mumford Conjecture
Congling QIU (邱聪灵), Princeton University
Abstract: Jinbo talked about the Pila-Zannier approach to Manin-Mumford conjecture, which works in the complex universal cover of an AV, and gets certain bounds on torsion points. I want to go back to Raynaud, whose proof involves a “p-adic universal cover” (in a sense), and essentially doesn’t need such bounds. I will review his proof, and later give another “p-adic universal cover” (in a sense) which is a perfectoid space, and finally give another proof of the Manin-Mumford conjecture. (This is inspired by Junyi Xie’s perfectoid approach to certain case of the dynamic Manin-Mumford conjecture.
No. SM-162 15/10/2016 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
Comptage de systèmes locaux l-adiques sur les courbes sur les corps finis
Hongjie YU (余红杰), Université Paris 7
Résumé: Drinfeld a publié en 1981 un article d’une seule page de compter les représentations l-adiques irréductibles du groupe fondamental d'une courbe sur un corps fini. Curieusement, le comptage de Drinfeld est semblable à celui du nombre de points d'une variété sur un corps fini. Par la correspondance de Langlands, démontrée par Drinfeld et Lafforgue, il est équivalent de compter des représentations automorphes. Ce comptage est donc en principe accessible par la formule des traces d'Arthur.
Dans cet exposé, je vais tout d'abord presenter la correspondence de Langlands pour GL(n) avec quelques "corollaires" directes. Ensuite, je vais expliquer comment utiliser la formule des traces d’Arthur pour calculer le nombre. Finalement, je expliquerai comment refaire ce comptage sans utiliser la formule des traces d’Arthur. Je vais limiter mon exposé aussi accessible que possible.
No. SM-161 01/10/2016 14:00 ~ 16:00 Amphithéâtre Léon Motchane, IHES
A gluing formula for the analytic torsion
Yeping ZHANG (张野平), Université Paris-Sud
Abstract: 161_Yeping ZHANG_abstract.pdf