Realizing Galois representations in abelian varieties by specialization

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.