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Surjectivity of Galois
representations in rational families of abelian varieties
Aaron Landesman, Ashvin A. Swaminathan, James Tao and Yujie XuAppendix: Davide Lombardo |
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Vol. 13 (2019), No. 5, 995–1038
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Abstract |
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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- 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 , there are infinitely many abelian varieties over with adelic Galois representation having image equal to all of . |
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Keywords
Galois representation, abelian variety, étale fundamental
group, large sieve, big monodromy, Hilbert irreducibility
theorem
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Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11G10, 11G30, 11N36, 11R32, 12E25
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Milestones
Received: 26 October 2017
Revised: 1 October 2018
Accepted: 22 February 2019
Published: 12 July 2019
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Authors |
| Aaron Landesman | |
| Department of Mathematics Stanford University CA United States |
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| Ashvin A. Swaminathan | |
| Department of Mathematics Princeton University NJ United States |
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| James Tao | |
| Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States |
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| Yujie Xu | |
| Department of Mathematics Harvard University Cambridge, MA United States |
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| Davide Lombardo | |
| Dipartimento di Matematica Università di Pisa Italy |
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