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.[-]

In their preprint about the Shafarevich conjecture for hypersurfaces on abelian varieties, Lawrence and Sawin prove a big monodromy theorem for families of hypersurfaces by reducing it to a similar result for Tannaka groups of perverse intersection complexes. A large part of their work is an intricate combinatorial argument about Hodge numbers, which is used to exclude that the Tannaka group acts via wedge powers of the standard representation of SL(n). We explain a simple geometric proof of the analogous result when hypersurfaces are replaced by subvarieties of high codimension; this is joint work in progress with Ariyan Javanpeykar, Christian Lehn and Marco Maculan.[-]