: April 2022 :
Langue: anglais
Résumé: Rank 2 Drinfeld modules can be regarded as analogue of elliptic curves in function field. We talk about an unlikely intersection problem that has been proved by Habegger for elliptic curves, i.e. there are finitely many unitary singular moduli. In his proof, some techniques from Diophantine geometry are used by Habegger. For Drinfeld modules, many of these techniques are missed. We will talk about an upper bound for Weil heights of the singular moduli of rank 2 Drinfeld modules and the techniques developed for Drinfeld modules so far.
: April 2022 :
Langue: anglais
Résumé: Locally symmetric spaces are generalizations of modular curves, and their cohomology plays an important role in the Langlands program. In this talk, I will first speak about vanishing conjectures and known results about the cohomology of locally symmetric spaces of a reductive group $G$ with mod $p$ coefficient after localizing at a maximal ideal of Spherical Hecke algebra of $G$ and after that, I will explain a sketch of my proof for the case $G = GL_2(F)$, where $F$ is a CM field.
: March 2022 :
Langue: anglais
Résumé: For the past decade, the formalism of perfectoid spaces has greatly reformed the modern number theory. One reason for introducing perfectoid spaces is to study rigid varieties without assuming that the rigid varieties come from algebraic geometry. The rigid varieties have many more points than algebraic varieties, and has very different topological nature. From this point view, perfectoid spaces for rigid geometry are the counterpart of affine spaces for algebraic geometry, and instead of analytic topology, we will need a finer topology, namely proétale topology. I will give a proof of finiteness of cohomology for proper varieties as an application.
: March 2022 :
Langue: anglais
Résumé: The mod p Langlands correspondence is completely known for the group GL2(Qp) by the work of Breuil, Colmez, etc. However, the situation becomes much more difficult when we replace Qp by a nontrivial finite extension. By a result of Emerton, the mod p Langlands correspondence for GL2(Qp) can be realized in the cohomology of modular curves, thus it is natural to look for a hypothetical correspondence for GL2 in the cohomology of Shimura curves and there have been many works on the study of the representations of GL2 coming from the cohomology of Shimura curves in the context of local-global compatibility. In this talk I will review some of the past results, then I will present a recent work of Breuil-Herzig-Hu-Morra-Schraen which computes the Gelfand-Kirillov dimensions of these representations.
: February 2022 :
Langue: anglais
Résumé: During the 1960s, Jennings, Golod, Shafarevich and Lazard introduced two sequences of integers a and c, closely related to a special filtration of a finitely generated pro-p group G, called Zassenhaus filtration. These sequences give the cardinality of G, and characterize its topology. Let us cite the famous Gocha's alternative: this is a condition on a and c, equivalent for G to be analytic, i.e. a Lie group over p-adic fields. Recently in 2016, Minac, Rogelstad and Tan inferred an explicit relation between the previoys sequences. This talk will review these results, enrich them in an isotypical context, and give examples.
: February 2022 :
Langue: anglais
Résumé: Nadel theorem ensures that the Baily-Borel compactification of a Shimura variety with sufficiently high level is Kobayashi hyperbolic. A conjecture of Lang predicts that a hyperbolic smooth projective variety defined over number field has at most finitely many raitonal points. For curves, Lang conjecture reduces to Mordell conjecture proved by Faltings. Lang conjecture remains open for surfaces. However, for adjoint abelian type Shimura varieties, Ullmo proved finiteness of integral points. He, together with Yafaev, also established an alternative principle for rational points on general Shimura varieties, which (roughly speaking) means Lang conjecture is either true or very false.
: January 2022 :
Langue: Français
Résumé: Soit $S$ une surface algébrique lisse. Le schéma de Hilbert $S^{[n]}$ paramétrant les sous-schémas artiniens de longueur $n$ est un objet géométrique intéressant qui fournit des exemples utiles de la géométrie algébrique. Par exemple, à l'aide des constructions de schémas de Hilbert, Beauville a construit des exemples de variétés hyper-kählériennes de dimensions supérieures, invalidant une conjecture de Bogomolov. Dans cette discussion, nous allons nous intéresser aux groupes cohomologiques de Betti et aux groupes de Chow des schémas de Hilbert de points sur une surface lisse, suivant les travaux de Göttsche et Soergel, et de de Cataldo et Migliorini.
: November 2021 :
Langue: Français
Résumé: Un groupe fini étant donné, il est légitime de se demander s'il est isomorphe au groupe de Galois d'une extension galoisienne du corps des rationnels: c'est le problème de Galois inverse. De Noether à nos jours, différentes méthodes ont été suggérées pour attaquer cette question. Après avoir donné un aperçu historique du problème, je présenterai une stratégie géométrique développée lors des dernières décennies -- consistant à approcher des points adéliques par des points rationnels sur certains espaces homogènes -- puis je présenterai un travail en cours avec Danny Neftin où nous donnons une réponse positive au problème de Galois inverse pour une nouvelle famille de groupes non résolubles.
: November 2021 :
Language: English
Abstract: In this talk we explain how the theory of condensed mathematics of Clausen and Scholze helps to treat the classical theory of locally analytic representations in a purely algebraic way, namely, as the theory of modules of an inverse system of distribution algebras. This phenomena was already observed by Schneider-Teitelbaum after restriction to admissible representations, and applied, for example, to compute extensions of principal series via Lie algebra cohomology. The advantage of the condensed approach is that the topology of the representations (and any reasonable topological space) are part of the ‘’algebraic data’’ of the condensed set. We manage to work in a derived category, in such a way that the classical comparison theorems of Lazard, Tamme, et. al., between continuous, locally analytic, and Lie algebra cohomology are deduced formally from the theory. This is joint work with Joaquín Rodrigues Jacinto.
: October 2021 :
Language: English
Abstract: When we study automorphic representations of a reductive Q-group G, sometimes we need G to be the generic fiber of some reductive Z-group scheme. If this holds, we say that G admits a Z-model. In SGA3, the theory of Chevalley groups tells us any split connected reductive Q-group has a unique Z-model up to Z-group isomorphism. For semisimple groups there are also some non-split examples. However, not all non-split semisimple Q-groups have Z-models. In his famous survey paper Groups over Z, Gross states two necessary and sufficient conditions for semisimple groups to admit Z-models, which are proved by Harder, and enumerates all the possibilities via the mass formula in some cases. In this talk, I will introduce these conditions and the mass formula, and then follow Gross’s route to construct Z-models for these non-split Q-groups, especially for anisotropic groups of exceptional types G2,F4.
: June 2021 :
Language: English
Abstract: Let $G$ be a pro-$p$-group, which admits a minimal presentation, with $d$ generators and $r$ relations. In $1964$, Golod and Shafarevich showed that if $G$ is a $p$-group, then it satisfies $d^2<4r$. The original proof of this result use a very subtle study of Poincaré Series. Poincaré Series gives also cohomological information on pro-$p$-groups. During the 60's, Lazard and Koch showed that a pro-$p$-group has cohomological dimension less than two if and only if its Poincaré Series verifies some equality. Between 1980 and 2000's, Anick and Labute, introduced a sufficient and easy condition on the relations of pro-$p$-group $G$, such that $G$ is of cohomological dimension less than two. Groups satisfying this sufficient condition are called mild. In this talk, we will present, more precisely, Poincaré series, cohomological consequences, and mild groups. If time permits, we will give some examples in an arithmetic context.
: June 2021 :
Language: English
Abstract:When deciding the existence of integer solutions of a system of polynomial equations with integer coefficients (defining a variety X), one could first reduce the problem modulo every integer N, which is equivalent to considering solutions in every Z_p (the p-adic integers). Similarly, we can consider rational solutions, by first looking at the candidate solutions “locally” in every p-adic field Q_p. Does the existence of "local" solutions in every Q_p give a "global" solution over Q (known as "local-global principle")? Sometimes yes (e.g Hasse-Minkowski theorem for quadratic forms), but not always. In fact, when local points ∏X(Q_p) which potentially come from global points X(Q) are paired with elements of the Brauer group Br(X) or other cohomology groups, they should satisfy certain restrictions, e.g. the exact sequence from Class Field Theory relating Br(Q) to Brauer groups of all the local Q_p, and this defines the Brauer-Manin obstruction. This obstruction is enough to detect the existence of rational points when X satisfies certain properties, e.g. being homogeneous spaces of some nice algebraic groups like tori. When there do exist rational points, sometimes we can simultaneously approximate Q_p-points for finitely many places p by a rational point, i.e. X(Q) is dense in ∏X(Q_p), known as weak approximation. Similarly, we look for obstructions when this doesn't hold: we hope that the closure of X(Q) in ∏X(Q_p) should be in some (closed) subset, e.g. the one defined by the Brauer-Manin obstruction. Now, instead of working over number fields, we want to generalize these results to function fields of complex curves or surfaces, where we can define the counterpart of local p-adic completions using valuations given by codimension 1 points.
: May 2021 :
Language: English
Abstract:Artin's conjecture states that Artin $L$-functions associated with non-trivial irreducible complex representations of Galois groups are analytic in the whole complex plane. For $2$-dimensional complex representations of Galois groups of totally real number fields, this conjecture is known to be true in many cases. The idea is to show that those Artin $L$-functions come from $L$-functions of cusp modular forms for which the analyticity is known. However, in some situations, one cannot directly obtain the existence of the associated modular forms, rather only so-called overconvergent $p$-adic modular forms. Hence a key step is to prove a classicality result showing that the $p$-adic modular forms we find are indeed classical modular forms, which can be established by the analytic continuation method (after Buzzard, Taylor, Kassaei, Pilloni...). In this talk, I will introduce some basic ideas around $p$-adic modular forms and their analytic continuation.
Link: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)
Password: e23J78
: April 2021 :
Language: English
Abstract:By Poisson summation formula, theta series of a positive definite quadratic lattice is a classical modular form. Such modularity has many applications, e.g Lagrange's four-square theorem. Generating series of special algebraic cycles on orthogonal or unitary Shimura varieties can be regarded as an arithmetic analogue of theta series. Kudla conjectured its modularity in general. For example, generating series of Heegner points on modular curves is known to be modular. We are interested in the modularity of Kudla-Rapoport divisors, on the RSZ-variant of unitary Shimura varieties. After some backgrounds, we will explain the recent work of [AW] about an almost modularity result on the hyperspecial level. Such almost modularity on integral models has many arithmetic applications. Time permitting, we will explain our work about almost modularity on certain parahoric levels with bad reductions.
Link: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)
Password: e23J78
: March 2021 :
Language: English
Abstract: In the last decade, the field of p-adic Hodge theory has advanced enormously, thanks to Scholze's development of perfectoid geometry, and Fargues-Fontaine's discovery of 'the fundamental curve'. In the first part of the talk, I will give an introduction to this field with an eye on the historical examples that motivated the development of (rational) p-adic comparison theorems for smooth proper rigid-analytic varieties, culminating in Scholze's proof of a de Rham comparison theorem, and Colmez-Niziol's proof of a semistable comparison theorem for such varieties. In the second part of the talk, I will focus on the p-adic Hodge theory of more general (i.e. including non-proper, e.g. Stein) smooth rigid-analytic varieties: the study of this subject has been undertaken in recent years mainly by Colmez-Dospinescu-Niziol, and it is motivated by the desire of finding a geometric incarnation of the (still widely conjectural) p-adic Langlands correspondence in the p-adic cohomology of local Shimura varieties. One difficulty here is that the relevant cohomology groups (such as the p-adic (pro-)étale, and de Rham ones) of non-proper rigid-analytic varieties are usually huge, and it becomes important to exploit the topological structure that they may carry in order to study them; but, in doing so, one quickly runs into topological issues, mainly due to the fact that the category of topological abelian groups is not abelian. I will explain how to overcome these issues, using the condensed and solid formalisms developed by Clausen-Scholze, and I will report on attempts of proving a general comparison theorem describing the geometric p-adic pro-étale cohomology in terms of de Rham data, for a large class of smooth rigid-analytic varieties defined over a p-adic field.
Link: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)
Password: e23J78
: February 2021 :
Language: English
Abstract: Research on computations of geometry invariants of curves is a hot topic in algebraic geometry. Computation on gonality of curves with large degree is a fundamental one. This is related to the syzygy resolutions of the curves via Green's gonality conjecture, claimed by Green and Lazarsfeld in 1986, that some conditions on gonality of a curve of large degree is equivalent to the vanishing conditions on Koszul cohomology groups. I will show how this equivalence is set up. There are further results like effective bounds on the degree of the curve, and generalizations to higher dimensional cases.
Link: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)
Password: e23J78
Une variété algébrique est dite rationnelle si elle est birationnellement équivalente à l'espace projectif. Le problème de décider quelles variétés sont rationnelles occupe les géomètres algébristes depuis le 19ème siècle et est encore, malgré de grands progrès, largement ouvert à l'heure actuelle. La théorie des jacobiennes intermédiaires joue un rôle tout particulier dans ces questions dans le cas des variétés de dimension 3, aussi bien sur les complexes (Clemens et Griffiths, dans les années 1970) que sur des corps non algébriquement clos (travaux en collaboration avec Olivier Benoist). L'exposé sera consacré au contexte et aux principales idées sous-jacentes au rôle des jacobiennes intermédiaires dans les questions de rationalité.
: December 2020 :
Language: English
Abstract: This talk is based on my thesis supervised by P.-H. Chaudouard. The conjecture of Guo-Jacquet is a promising generalisation to higher dimensions of Waldspurger’s well-known theorem on the relation between toric periods and central values of automorphic L-functions for GL(2). However, we are faced with divergent integrals when applying the relative trace formula approach. In this talk, after briefly introducing the background, we shall focus on an infinitesimal variant of this problem. Concretely, we shall explain global and local trace formulae for infinitesimal symmetric spaces of Guo-Jacquet. To compare regular semi-simple terms, we shall present the weighted fundamental lemma and certain identities between Fourier transforms of local weighted orbital integrals.
Link: https://zoom.us/j/92227425559?pwd=VGRFam5mOEtBTmFXYWwzalRBV2FRUT09 (ID de réunion: 922 2742 5559)
Password: e23J78
: November 2020 :
Language: English
Abstract: https://drive.google.com/file/d/1wIOR4epF5YSYmpWXWXh6NxRChwUpWTsD/view
Link: https://zoom.us/j/92227425559?pwd=VGRFam5mOEtBTmFXYWwzalRBV2FRUT09 (ID de réunion: 922 2742 5559)
Password: e23J78
: November 2020 :
Language: English
Abstract: https://drive.google.com/file/d/1n0hE-NdUeKnBOsAmGL4SPeW7LspvyX95/view
Link: https://bbb.math.univ-paris13.fr/b/bou-swj-b9t-nib