A motivic formula for the L-function of an abelian variety over a function field

Abstract

Let $A$ be an abelian variety over the function field of a smooth projective curve $C$ over an algebraically closed field $k$. We compute the $l$-adic cohomology groups of $C$ with coefficients in the locally constant sheaf associated to $H^1(\bar{A},{\mathbf{Q}}_l)$ in terms of arithmetico-geometric invariants of $A$. We apply this, when $k$ is the algebraic closure of a finite field, to a motivic computation of the $L$-function of $A$.

Substantially revised. In particular proofs in Section 2 are simplified and clarified, and the motivation for Section 7 is more detailed

Cited by in corpus (1)

• On the $\mu$-invariants of abelian varieties over function fields of positive characteristic