#### Vol. 13, No. 5, 2019

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Surjectivity of Galois representations in rational families of abelian varieties

### Appendix: Davide Lombardo

Vol. 13 (2019), No. 5, 995–1038
##### Abstract

In this article, we show that for any nonisotrivial 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 $ℚ$ with adelic Galois representation having image equal to all of ${GSp}_{2g}\left(\stackrel{̂}{ℤ}\right)$.

##### Keywords
Galois representation, abelian variety, étale fundamental group, large sieve, big monodromy, Hilbert irreducibility theorem
##### Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11G10, 11G30, 11N36, 11R32, 12E25