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arXiv · 1608.05371

Surjectivity of Galois Representations in Rational Families of Abelian Varieties

Aaron Landesman·Ashvin Swaminathan·James Tao·Yujie Xu
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Abstract

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation having image equal to all of $\operatorname{GSp}_{2g}(\widehat{\mathbb Z})$.

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Aaron Landesman, Ashvin Swaminathan, James Tao, Yujie Xu. 2016-08-18. Surjectivity of Galois Representations in Rational Families of Abelian Varieties. https://doi.org/10.2140/ant.2019.13.995

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