SM Exposés 2021-2022

• Langue: english

• Résumé de l'exposé: In this talk, we will show  a general method to establish tame and norm relations for special cycles in Shimura varieties. We use unitary cycles in odd orthogonal Shimura varieties as a guiding example. We also list some other examples like GSp(4) Shimura varieties  and unitary Shimura varieties. Such special cycles with desired distribution relations is a higher dimensional generalization of Heegner points. It can be applied to construct an Euler system to do arithmetic applications etc.

• Langue: english

• Résumé de l'exposé: In 1976, Quillen and Suslin (independently) solved Serre's problem, which claims that for a field $k$, any vector bundle over affine space $\mathbb{A}^{n}_{k}$ is trivial. A generalization of Serre's problem is known as the Bass-Quillen conjecture: for a regular ring $R$, every vector bundle over affine space $\mathbb{A}^{n}_{R}$ descends to $R$. This conjecture was established in many cases, notably for unramified regular rings. Generally, one could expect the same statement for generically trivial torsors under totally isotropic reductive group schemes.

In this talk, we will review the proof of the Bass-Quillen conjecture in unramified case. If time permits, we will sketch recent progress on the Bass-Quillen conjecture for torsors by Fedorov and Česnavičius.

• Langue: to be defined

• Résumé de l'exposé: Strong approximation theorems are ubiquitous in arithmetic geometry and have been broadly studied over global fields. But what happens if the base field is replaced by the function field of a complex algebraic curve? This question has been being tackled since the last decade, using deformation theory of algebraic curves. In this presentation, we shall give some generalities on strong approximation for varieties over $\mathbf{C}(t)$, and discuss the remanence of this property on the complementary of any closed subset of codimension 2.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: français

• Résumé de l'exposé: 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.

• Langue: français

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.