The classical Brauer-Siegel theorem can be seen as one of the first instances of description of asymptotical arithmetic: it states that, for a family of number fields $K_i$, under mild conditions (e.g. bounded degree), the product of the regulator by the class number behaves asymptotically like the square root of the discriminant.
This can be reformulated as saying that the Brauer-Siegel ratio log($hR$)/ log$\sqrt{D}$ has limit 1.
Even if some of the fundamental problems like the existence or non-existence of Siegel zeroes remains unsolved, several generalisations and analog have been developed: Tsfasman-Vladuts, Kunyavskii-Tsfasman, Lebacque-Zykin, Hindry-Pacheco and lately Griffon. These analogues deal with number fields for which the limit is different from 1 or with elliptic curves and abelian varieties either for a fixed variety and varying field or over a fixed field with a family of varieties.[-]

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.[-]

The Hasse-Weil zeta function of a regular proper flat scheme over the integers is expected to extend meromorphically to the whole complex plane and satisfy a functional equation. The local epsilon factors of vanishing cycles are the local factors of the constant term in the functional equation. For their absolute values, Bloch proposed a conjecture, called Bloch's conductor formula, which describes them in terms of the Euler characteristics of a certain (complex of) coherent sheaf. In this talk, under the assumption that the non-smooth locus is isolated and that the residue characteristic is odd, I explain that the coherent sheaf appearing in the Bloch's conjecture is naturally endowed with a quadratic form and I would like to propose a conjecture that describes the local epsilon factors themselves in terms of the quadratic form. The conjecture holds true in the following cases: 1) for non-degenerate quadratic singularities, 2) for finite extensions of local fields, or 3) in the positive characteristic case.[-]

(joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou)
We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal tran-scendence degree.
The proof uses a strong form of the Lang-Weil estimates and, as key ingredi-ent to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.[-]