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

Let $X$ be a variety over a finite field. Given an order $R$ in a semi-simple algebra over the rationals and a constructible étale 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-étale 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.