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arXiv · 1910.14174

Families of abelian varieties and large Galois images

David Zywina
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Abstract

Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $ρ_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $ρ_A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a nonempty open subvariety $U\subseteq \mathbb{P}^n_K$. We show that away from a set of density $0$, the image of $ρ_A$ will be very large; more precisely, it will have uniformly bounded index in a group obtained from the family of abelian varieties. This generalizes earlier results which assumed that the family of abelian varieties have "big monodromy". We also give a version for a family of abelian varieties with a more general base.

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David Zywina. 2019-10-30. Families of abelian varieties and large Galois images. https://arxiv.org/abs/1910.14174

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