Mathematical Research Letters

Volume 20 (2013)

Number 1

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

Pages: 1 – 17



Sara Arias-De-Reyna (Faculté des Sciences, de la Technologie et de la Communication, University of Luxembourg, Coudenhove-Kalergi, Luxembourg)

Christian Kappen (Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Essen, Germany)


Given a natural number $n\geq 1$ 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\[\rho_{A,\ell}:\,G_F\rightarrow \mathop{\mathrm{GSp}}(A[\ell](\overline{F}))\]is surjective and everywhere tamely ramified. In particular, we realize ${\mathop{\mathrm{GSp}}}_{2n}(\mathbb{F}_\ell)$ 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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