Weil-\'etale cohomology and the equivariant Tamagawa number conjecture for constructible sheaves in characteristic $p$

Abstract

Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible \'etale sheaf $F$ of $R$-modules over $X$, one can consider a natural non-commutative $L$-function associated with $F$. We prove a special value formula at negative integers for this $L$-function, expressed in terms of Weil-\'etale cohomology; this is a geometric analogue of, and implies, the equivariant Tamagawa number conjecture for an Artin motive and its negative twists over a global function field. It also generalizes the results of Lichtenbaum and Geisser on special values at negative integers for zeta functions of varieties, and the work of Burns--Kakde in the case of non-commutative L-functions coming from a Galois cover of varieties.