SM Exposés 2020-2021

• Langue:français

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: 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.

• Langue: anglais

• Résumé de l'exposé: Faltings' approach in p-adic Hodge theory can be schematically divided into two main steps: firstly, a local reduction of the computation of the p-adic étale cohomology of a smooth variety over a p-adic local field to a Galois cohomology computation and then, the establishment of a link between the latter and differential forms. These relations are organized through Faltings ringed topos whose definition relies on the choice of an integral model of the variety, and whose good properties depend on the (logarithmic) smoothness of this model. Scholze's generalization for rigid analytic varieties has the advantage of depending only on the variety (i.e. the generic fibre). Inspired by Deligne's approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings' approach to any integral model, i.e. without any smoothness assumption. An essential ingredient of our proof is a descent result of perfectoid algebras in the arc-topology due to Bhatt and Scholze. As an application of our cohomological descent, using a variant of de Jong's alteration theorem for morphisms of schemes, we generalize Faltings' main p-adic comparison theorem to any proper and finitely presented morphism of coherent schemes over an absolute integral closure of Z_p (without any assumption of smoothness) for torsion étale sheaves (not necessarily finite locally constant).

• Lien pour l'exposé: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)

• Mot de passe d'accès: e23J78

• Langue: anglais

• Résumé de l'exposé: 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.

• Lien pour l'exposé: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)

• Mot de passe d'accès: e23J78

• Langue: anglais

• Résumé de l'exposé: 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.

• Lien pour l'exposé: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)

• Mot de passe d'accès: e23J78

• Langue: anglais

• Résumé de l'exposé: 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.

• Lien pour l'exposé: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)

• Mot de passe d'accès: e23J78

• Langue: anglais

• Résumé de l'exposé: 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.

• Lien pour l'exposé: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)

• Mot de passe d'accès: e23J78

• Langue: anglais

• Résumé de l'exposé: Résumé en PDF/File of the abstract
  Consider a polarised abelian scheme of finite type over a complete valued field k. The set of rational points of the base has the structure of a k-analytic space and acquires a natural topology as such.
  
  Question: For a positive integer d, consider the set R_d of those rational points of the base whose corresponding fiber contains an abelian subvariety of dimension d. Can we give a natural criterion for the density of this set R_d in the set of all rational points of the base?
  
  In this talk I will positively answer this question in the case when k is either the field of real or of complex numbers. Indeed, over the complex numbers this was proved by Colombo and Pirola using Hodge theory; my goal has been to adapt their proof to the real numbers.
  
  Surprisingly, the resulting criterion is exactly the same as in the complex setting. In some cases the criterion can be satisfied, yielding the following results over the complex as well as over the real numbers: abelian varieties that contain a d-dimensional abelian subvariety are dense in the moduli space of principally polarised abelian varieties of dimension g greater or equal than d; for small values of d, algebraic curves that admit a non-trivial morphism to an abelian variety of dimension d are dense in the moduli space of genus g \geq 3 curves; and plane curves that map non-trivially to elliptic curves are dense in the moduli space of degree d smooth plane curves.
  
  To explain these results, I shall start with short introduction to real algebraic geometry. For example, by GAGA we have an equivalence of categories between complex projective varieties and complex projective manifolds; given a projective variety over the complex numbers, what does a descent datum to move from the complex to the real numbers correspond to in the category of complex manifolds? Then I will recall the most important expects of Hodge theory, and motive them by the famous Green-Voisin Density Criterion: a criterion for the density of the Noether-Lefschetz locus. I will show how Colombo and Pirola adapted the Green-Voisin Density Criterion to the case of a polarised variation of Hodge structure of weight one (which is nothing but a polarised family of complex abelian varieties!). Finally, I shall combine these theories (real algebraic geometry and Hodge theory) to answer Question 1 over the real numbers. If time permits, I shall sketch how satisfy the density criterion and provide applications.

• Lien pour l'exposé: https://zoom.us/j/6717858447?pwd=TFdIWm4zTG1LR3pqMStxRm0zd1R3QT09 (ID de réunion: 671 785 8447)

• Mot de passe d'accès: e23J78

Notes de l'exposé: https://drive.google.com/file/d/1fhzJ2dI1rDJtK_davVYHK644dxcFpOoH/view?usp=sharing

• Langue: anglais

• Résumé de l'exposé: 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.

• Lien pour l'exposé: https://zoom.us/j/92227425559?pwd=VGRFam5mOEtBTmFXYWwzalRBV2FRUT09 (ID de réunion: 922 2742 5559)

• Mot de passe d'accès: e23J78

• Langue: anglais

• Résumé de l'exposé: Résumé de l'exposé / Abstract of the presentation

• Lien pour l'exposé: https://zoom.us/j/92227425559?pwd=VGRFam5mOEtBTmFXYWwzalRBV2FRUT09 (ID de réunion: 922 2742 5559)

• Mot de passe d'accès: e23J78

• Langue: anglais

• Résumé de l'exposé: Résumé de l'exposé / Abstract of the presentation

• Lien pour l'exposé: https://bbb.math.univ-paris13.fr/b/bou-swj-b9t-nib

• Mot de passe d'accès: 625562