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 ≥ 3, there are infinitely many abelian varieties over Q with adelic Galois representation having image equal to all of GSp 2g ( Z).

Constructing hyperelliptic curves with surjective Galois representations

“…This approach has been successfully used in a number of works, including [Zar79], [SZ05], [Hal08], [Zar10], [Hal11], [AdRV11], [AdRK13], [AdRAK + 15], which realise GSp 2g (F ℓ ) for every (odd) prime ℓ using Jacobians of hyperelliptic curves, and show that one curve often realises GSp 2g (F ℓ ) for all sufficiently large ℓ. More recently [LSTX16] gave a non-constructive proof that many hyperelliptic curves realise GSp 2g (F ℓ ) for all odd primes ℓ, and [Die02], [Zyw15], [ALS16] who exhibited explicit curves of genus 2 and 3 with this property. There has also been numerical work [AdRAK + 16], [ALS16] investigating the Galois images of Jacobians of curves.…”