Abstracts for Conference in Memory of Yuri Manin

By Deligne's Hodge theory, the integral cohomology groups $H^n(\mathcal{X}^h,\mathbf{Z})$ of the $\mathbf{C}$-analytification $\mathcal{X}^h$ of a separated scheme $\mathcal{X}$ of finite type over $\mathbf{C}$ are provided with a mixed Hodge structure, functorial in $\mathcal{X}$. Given a non-Archimedean field $K$ isomorphic to the field of Laurent power series $\mathbf{C}((z))$, there is a functor $\mathcal{X}\mapsto \mathcal{X}^{\text{an}}_K$ that takes $\mathcal{X}$ to the non-Archimedean $K$-analytification of $\mathcal{X}_K = \mathcal{X}\otimes_{\mathbf{C}} K$.

The fundamental property of zeta functions and L-functions is that their meromorphic continuations provide a lot of information about the corresponding objects. Complex values of $s$ occur as a technical tool, with little direct arithmetic-geometric meaning. In the refined theory, $1/n^s$ are replaced by certain $q,t,a$-series, which are invariants of Lens Spaces $L(n,1)$ directly related to Elliptic Hall

The importance of the Brauer group for arithmetic geometry was highlighted by Manin in his celebrated 1970 ICM address. In this talk I will discuss the structure of the Brauer group of an abelian variety $A$ over an algebraically closed field of characteristic $p$ focusing on the $p$-primary torsion, the key part of which is a certain quasi-algebraic unipotent group $U_A$. I will present results on the dimension and the $p$-exponent of $U_A$ based on the classical Manin-Dieudonné theory, leading to the determination of $U_A$ up to isogeny for abelian varieties $A$ of small dimension.