Descent properties for an abelian variety with extended Galois representation

TL;DR

This paper investigates when abelian varieties defined over a Galois extension can be descended to the base field, focusing on the extension of Galois representations and specific conditions for Type I varieties.

Contribution

It establishes criteria under which the extension of Galois representations implies descent for abelian varieties of Type I, with conditions on endomorphisms and the nature of the base field.

Findings

Extension of Galois representations implies descent for Type I abelian varieties.

Additional conditions on endomorphisms are necessary for the converse.

Results apply over number fields and certain function fields.

Abstract

Let $K$ be a field, $L$ a finite Galois extension of $K$, and $X$ an abelian variety defined over $L$. If $X$ is isogenous over $L$ to an abelian variety defined over $K$, then the $\ell$-adic Galois representations associated to $X$ extend to representations $\bar{\rho}_{\ell,X}:\mathrm{Gal}(\bar{L}/K)\to\mathrm{Aut}(V_\ell X)$ for every prime $\ell$. This paper aims to show that the converse is true for abelian varieties of Type I, with some supplementary conditions needed on the endomorphisms of $X$, when $L$ is either a number field or a function field of prime characteristic different from $2$.