SM Exposés 2018-2019

Abstract : Understanding the density and the distribution of rational points on an algebraic variety is usually a very subtle problem, to which it is generally believed that there is no uniform approach.

In this talk we try to explain how sieve methods coming from analytic number theory naturally complements geometric and cohomological arguments.

More precisely, we will see how Brun’s affine combinatorial sieve developed by Sarnak et al and Ekedahl-Poonen’s geometric sieve can (potentially) be matched with the fibration method in studying strong approximation of rational points in the adelic space.

This is a joint project with Yang Cao.

Abstract : In order to study some arithmetic properties over finitely generated fields, Atsushi Moriwaki (森脇 淳) introduced a height function on projective varieties over these fields. This kind of height functions is related to some arithmetic line bundles over the normal model of the finitely generated field. In this talk, we will introduce the definition and some properties of this height function ,especially the condition of its Northcott’s property. We will also introduce some applications of this height function in several branches of arithmetic geometry.

Notes of the talk

Abstract : Let S be a PEL Shimura variety associated to a reductive group G. Every algebraic representation V of G gives rise to a mixed sheaf µ(V) on S (either in the Hodge-theoretical or l-adic sense). Theorems of Wildeshaus show that the understanding of the weight filtration on the cohomology of µ(V) can lead to the construction of Chow motives associated to cuspidal automorphic representations of G. The aim of the talk is to introduce this circle of ideas and, time permitting, to explain the results obtained for G=GSp_4/F (F a totally real field): we prove that for the associated varieties, the weight filtration is controlled by a representation-theoretic invariant, called corank. This allows to construct the desired motives in many cases.

Abstract : Let G be a semisimple group scheme over an algebrically closed field k of positive characteristic. In 1980, Jantzen has proved that under certain assumptions of genericity, every Weyl module V(λ) ≅ H^n(w_0 · λ) of G has a « p-Weyl-filtration », i.e. a filtration whose quotients are of the form L(µ^0)\otimes V(µ^1)^{(1)}. If G=SL_3, Jantzen has proved that this kind of filtration exists for all V(λ). It is interesting to ask whether or not every H^i(λ) has a similar filtration whose quotients are of the form L(µ^0)\otimes H^i(µ^1)^{(1)}. This, however, is not true even in the case of SL_3. In this talk, I will define a slightly cruder filtration and prove its existence for all H^i(λ) if G=SL_3.

Abstract : Let k be a finitely generated field of positive characteristic p, X a smooth geometrically connected k-variety and f:Y---->X a smooth proper morphism.

We study how the geometry and the arithmetic of fiber Y_x of f at x vary with x in X. In particular we will study the variation of the Néron-Severi group. To do this, we associate to f:Y---->X various local systems (an l-adic lisse sheaf, an F-isocrystal and an overconvergent F-isocrystal ) and to each of them various (algebraic) groups. The interplay between the Néron-Severi group and these algebrac groups will be the main topic of this talk. Time permitting, I will talk about further applications to abelian varieties of the study of these algebraic groups.

Abstract : A really nice geometrical result from Kontsevich yields a recursive formula allowing us to compute the number of rational curves going through N points in general position in the projective space (through quantum cohomology and the use of the compactification of the moduli space of rational curves by stable maps). We shall first review this fundamental idea, before exploring how quantum K-theory generalises this problem, and exposing some results on the small quantum K-theory of some homogeneous varieties.

Abstract : In this talk, we survey the recent development of p-adic cohomology theories, namely the constructions of $A_{inf}$-cohomology, Breuil-Kisin cohomology and prismatic cohomology which is considered as the « right » cohomology . We will point out how these new cohomology theories are related to p-adic etale cohomology and crystalline cohomology.

Abstract : Étant donné un groupe réductif $G$, on s'intéresse à la catégorie $\mathrm{Perv}_G(\mathfrak{g}^{\mathrm{nil}})$ des faisceaux pervers équivariants sur le cône nilpotent de son algèbre de Lie.

G. Lusztig, en généralisant une construction de T. Springer, a établi une paramétrisation des objets simples en termes de représentations de certains groupes de Weyl, avant de répondre à la même question quand $\mathfrak{g}$ est munie d'une $\mathbf{Z}$-graduation, en termes de certaines algèbres de Hecke affines dégénérées.

Dans cet exposé, je vais expliquer ces constructions. Je parlerai également de travaux récents qui généralisent ces résultats dans le contexte où $\frak{g}$ est munie d'une graduation cyclique.

Abstract : The Selmer group of a p-adic Galois representation r of the absolute Galois group of Q that is ordinary at p contains important arithmetic information of r, for example the Mordell-Weil rank of E(Q) when r comes from an elliptic curve E/Q. In this talk, we will consider Galois representation r which is certain symmetric power of 2-dimensional Galois representation attached to a p-ordinary modular form. For the Z_p cyclotomic extension Q'/Q, we will establish a relation between the Fitting ideal of the Selmer group of r restricted to Q' and the Fitting ideal of the Selmer group of r, under some conditions on the image of the residual Galois representation of r.

Abstract : One technique often applied to prove modularity lifting theorems is the Taylor-Wiles-Kisin method, which, roughly speaking, is a process of patching modules or mophims by passing to the limit. We shall discuss some basic Galois deformation theory first and then investigate one version of the patching method. We will also make a comparison with some other versions of the patching argument.

Abstract : I will introduce local and global epsilon factors, and Gabber-Katzextensions, and I will then explain how the latter yield a cohomological definition "à la Laumon" of local epsilon factors. When A is a finite field, we recover the local epsilon factors defined by Langlands and Deligne by taking the trace of the Frobenius endomorphism.

Abstract : For a reductive group G over a semilocal Dedekind ring R with the fraction field K, we prove any rationally trivial G-torsor is trivial, that is, the map of pointed sets

H^1(G, R)--------->H^1(K, R)

is injective. This paper generalizes Nisnevich's result of the Grothendieck--Serre conjecture over discrete valuation rings.