Abelian varieties over number fields, tame ramification and big Galois image

TL;DR

This paper proves the existence of abelian varieties over certain number field extensions with large Galois images, tamely ramified at all primes above a large enough prime f, and realizes symplectic groups as Galois groups over number fields.

Contribution

It constructs abelian varieties with surjective f-torsion Galois representations that are tamely ramified and unramified above f, extending Galois realizations of symplectic groups.

Findings

Existence of abelian varieties with large Galois images over number fields.

Construction of tame Galois extensions with prescribed symplectic Galois groups.

Realization of GSp_{2n}(\u0000f_ ext{finite}) as Galois groups over number fields.

Abstract

Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting \ell-torsion representation \rho_{A,\ell} from G_F to GSp(A[\ell](\bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(\mathbb{F}_\ell) as the Galois group of a finite tame extension of number fields F'/F such that F is unramified above \ell.