Vol. 13, No. 5, 2019

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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
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 3, there are infinitely many abelian varieties over with adelic Galois representation having image equal to all of GSp2g( ̂).

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