Surjectivity of Galois Representations in Rational Families of Abelian Varieties

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 $g\ge 3$, there are infinitely many abelian varieties over $\mathbb{Q}$ with adelic Galois representation having image equal to all of ${\mathrm{GSp}}_{2g}(\hat{\mathbb{Z}})$.

arXiv comment41 pages

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Cited by in corpus (6)

• Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians
• Lifting Subgroups of Symplectic Groups over $\mathbb{Z}/\ell \mathbb{Z}$
• A Lefschetz theorem for intersections with projective varieties
• The $T$-adic Galois representation is surjective for a positive density of Drinfeld modules
• Arithmetic rank bounds for abelian varieties over function fields
• On the $L$-polynomials of curves over finite fields