Realizing Galois representations in abelian varieties by specialization

Abstract: We present a strategy for constructing abelian varieties J/K which realize a given rational Galois representation $\rho : G_K \to GL_n(Q)$, i.e. such that $\rho$ is a subrep of $J(\Kbar)\otimes \Q$. When $\rho$ is the trivial rep., then $J/K$ realizes $\rho$ if and only if $J(K)$ is of rank at least $n$, and such families are usually constructed by the specialization method pioneered by Neron. Our strategy consists in taking an already existing construction of abelian varieties with large rank and, provided there is enough symmetry, twisting the construction to obtain non-trivial Galois actions on the points. After twisting, we use a simple generalization of the classical Neron specialization theorem (from trivial reps. to non-trivial reps.) We apply this procedure to a construction of Mestre and Shioda to prove the following: Given a representation $\rho: G_K \to GL_n(\Q)$, there exist infinitely many absolutely simple absolutely abelian varieties $J/K$ (which are in fact Jacobians of hyperelliptic curves) such that $\rho$ is a subrep. of the $G_K$ rep on $J(\Kbar) \otimes_{\Z} \Q$.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar