Realizing Galois representations in abelian varieties by specialization

TL;DR

This paper demonstrates how to realize Galois representations within abelian varieties over certain fields, providing explicit constructions and extending classical specialization techniques to produce abelian varieties with desired Galois actions.

Contribution

It introduces a method to construct abelian varieties of small dimension that realize given Galois representations, generalizing Néron's Specialization Theorem and applying it to various curve constructions.

Findings

Existence of infinitely many simple abelian varieties realizing given Galois representations

Extension of specialization techniques to produce curves with points of large degree

Construction of abelian varieties over Hilbertian fields for large dimensions

Abstract

We give some positive answers to the following problem: Given a field $K$ and a continuous Galois representation $\rho:G_K \to GL_n(\mathbf{Q})$, construct an abelian variety $J/K$ of small dimension such that $\rho$ is a sub-representation of the natural $G_K$-representation on $J(\bar{K}) \otimes_{\mathbf{Z}} \mathbf{Q}$. We prove that if $K$ is Hilbertian of characteristic different from $2$, then for any sufficiently large integer $g$ (depending on $\rho$) we can find infinitely many absolutely simple $g$-dimensional abelian varieties which realize $\rho$. We outline also a method of twisting a given symmetric construction of curves with many rational points to instead produce curves with closed points of large degree, and in this context we give a unified treatment of constructions of Mestre--Shioda and Liu--Lorenzini. The main results are obtained by applying a natural generalization of N\'eron's Specialization Theorem.