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

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

2012-04-24

arxiv.org/abs/1204.5441v1

Mathematics

Number Theory

15 pages

Scientific paper

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.

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