Character formula for Weil representations in terms of Frobenius traces

Abstract

It is known that the étale cohomology of a potentially good abelian variety over a local field $K$ is determined by its Euler factors over the extensions of $K$. We extend this to all abelian varieties, show that it is enough to take extensions where $A$ is semistable, and give a uniform version over $p$-adic fields where the extensions are the same for all abelian varieties of a given dimension. The results are explicit, and apply to a wide class of Weil–Deligne representations.