Surjectivity of Galois representations in rational families of abelian varieties

Landesman, Aaron; Swaminathan, Ashvin; Tao, James; Xu, Yujie

doi:10.2140/ant.2019.13.995

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Surjectivity of Galois representations in rational families of abelian varieties

Aaron Landesman, Ashvin A. Swaminathan, James Tao and Yujie Xu

Appendix: Davide Lombardo

Vol. 13 (2019), No. 5, 995–1038

DOI: 10.2140/ant.2019.13.995

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

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

Milestones

Received: 26 October 2017

Revised: 1 October 2018

Accepted: 22 February 2019

Published: 12 July 2019

Authors

Aaron Landesman
Department of Mathematics Stanford University CA United States

Ashvin A. Swaminathan
Department of Mathematics Princeton University NJ United States

James Tao
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

Yujie Xu
Department of Mathematics Harvard University Cambridge, MA United States

Davide Lombardo
Dipartimento di Matematica Università di Pisa Italy

Vol. 13, No. 5, 2019