Euler factors determine local Weil representations

TL;DR

This paper demonstrates that local Weil representations over a local field are uniquely determined by their Euler factors across extensions, providing explicit constructions and applications to elliptic and genus 2 curves.

Contribution

It introduces an explicit method to determine Weil representations from Euler factors and applies it to construct abelian varieties with specific rank properties.

Findings

Weil representations are determined by Euler factors over extensions.

Explicit construction method for local Weil representations.

Application to abelian varieties with constrained quadratic twists.

Abstract

We show that a Frobenius-semisimple Weil representation over a local field K is determined by its Euler factors over the extensions of K. The construction is explicit, and we illustrate it for l-adic representations attached to elliptic and genus 2 curves. As an application, we construct an absolutely simple 2-dimensional abelian variety over Q all of whose quadratic twists must have positive rank, according to the Birch-Swinnerton-Dyer conjecture.