Character formula for Weil representations in terms of Frobenius traces

TL;DR

This paper provides explicit formulas for Weil representations using Frobenius traces, extending known results to all abelian varieties and offering uniform descriptions over p-adic fields.

Contribution

It generalizes the character formula for Weil representations to all abelian varieties, including semistable cases, with uniform results over p-adic fields.

Findings

Explicit character formulas for Weil representations in terms of Frobenius traces.

Extension of known results to all abelian varieties, not just potentially good ones.

Uniform descriptions applicable across p-adic fields for varieties of fixed dimension.

Abstract

It is known that the etale 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.