Weil-étale cohomology of arithmetic schemes

by Alexey Beshenov (CIMAT)

Thursday 13 May 2021, 12:30 → 14:30 Europe/Moscow

818-1526-4739 (Zoom)

Let $X$ be an arithmetic scheme, by which we will mean that it is separated and of finite type over $\mathrm{Spec}\,\mathbb{Z}$. The corresponding zeta function $\zeta (X,s)$ is defined via the usual Euler product over the closed points of $X$, and for an integer $n$, the special value $\zeta^*(X,n)$ is the leading coefficient of the Taylor series of $\zeta (X,s)$ around s=n (assuming analytic continuation). Several conjectures, of varying generality, express $\zeta^* (X,n)$ in terms of arithmetic invariants attached to $X$ (such as algebraic $K$-theory, motivic cohomology, and regulators).

Lichtenbaum conjectured the existence of Weil-étale cohomology groups $H^i_{W,c} (X, \mathbb{Z}(n))$ that encode the special value of $\zeta^* (X,n)$. Thanks to the subsequent work of Geisser, Weil-étale cohomology is rather well-understood for varieties over finite fields $X/\mathbb{F}_q$, and the case of mixed characteristic was later considered by  Morin and Flach. The construction is based on étale motivic cohomology of $X$ and arithmetic duality.

I will talk about the general context behind Lichtenbaum's Weil-étale program, known results, particular examples, my contributions, and some open problems.

Slides Video