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

TL;DR

This paper computes the $l$-adic cohomology of a curve with coefficients related to an abelian variety and uses this to derive a motivic formula for the $L$-function of the variety over a function field.

Contribution

It provides a new motivic formula for the $L$-function of an abelian variety over a function field, linking cohomological invariants to arithmetico-geometric data.

Findings

Explicit computation of $l$-adic cohomology groups in terms of invariants.

Derivation of a motivic formula for the $L$-function.

Application to varieties over algebraic closures of finite fields.

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 $A$.