We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their $L$-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be classical, the strategy consists in first realizing them as exponential motives and computing their Hodge numbers by means of the irregular Hodge filtration. We show that all Hodge numbers are either zero or one, which implies potential automorphicity thanks to recent results of Patrikis and Taylor. The first talk will be concerned with the arithmetic aspects and in the second one we will present the Hodge theoretic computations. Joint work with Claude Sabbah and Jeng-Daw Yu.[-]

(joint work with G. Wüstholz) Roughly, $1$-dimensional periods are the complex numbers obtained by integrating a differential form on an algebraic curve over $\bar{\mathbf{Q}}$ over a suitable domain of integration. One of the alternative characterisations is as periods of Deligne $1$-motives.
We clear up the linear relations between these numbers, proving Kontsevich's version of the period conjecture for $1$-dimensional periods. In particular, a $1$-dimensional period is shown to be algebraic if and only if it is of the form $\int_\gamma (\phi+df)$ with $\int_\gamma\phi=0$. We also get formulas for the spaces of periods of a given $1$-motive, generalising Baker's theorem on logarithms of algebraic numbers.
The proof is based on a version of Wüstholz's analytic subgroup theorem for $1$-motives.[-]

​We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Although other geometric proofs of this result are known, our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow, and using them to quickly solve the problem by invoking the irreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs. Joint work with Davesh Maulik.[-]

Given a smooth scheme $X$ over the ring of integers of a $p$-adic field, we introduce the notion of a relative Breuil-Kisin-Fargues module $M$ on $X$. Each such $M$ simultaneously encodes the data of a lisse étale sheaf, a module with flat connection, and a crystal, whose cohomologies are then intertwined by a relative form of the $A_{inf}$ cohomology introduced in "Integral $p$-adic Hodge theory" by Bhatt-M-Scholze. They are moreover closely related to other work in relative $p$-adic Hodge theory, notably Faltings small generalised representations and his relative Fontaine Lafaille theory. Joint with Takeshi Tsuji.[-]

François Charles est professeur à l'Université Paris-Sud Orsay. Après une thèse soutenue en 2010 à l'Université Pierre et Marie Curie sous la direction de Claire Voisin, il a travaillé au CNRS et au MIT. Il est spécialiste de géométrie algébrique et de géométrie arithmétique, et s'intéresse en particulier à la géométrie des cycles algébriques. Il a reçu le prix Peccot en 2014.
Deux questions soufflées par Claire Voisin, sa directrice de thèse, lui ont été posées : s'il ressent le poids des incitations à publier (a priori les jeunes y sont plus soumis) et une deuxième concernant l'évolution récente de la géométrie algébrique (qui peut poser problème, avec « une certaine tendance à l'éparpillement »).[-]