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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Constructing hyperelliptic curves with surjective Galois representations

Authors: Samuele Anni and Vladimir Dokchitser
Journal: Trans. Amer. Math. Soc. 373 (2020), 1477-1500
MSC (2010): Primary 11F80; Secondary 12F12, 11G10, 11G30
Published electronically: November 5, 2019
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Abstract: In this paper we show how to explicitly write down equations of hyperelliptic curves over $ \mathbb{Q}$ such that for all odd primes $ \ell $ the image of the mod $ \ell $ Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the $ \ell $-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod $ \ell $ Galois representations.

The main result of the paper is the following. Suppose $ n=2g+2$ is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than $ n$ (this hypothesis is expected to hold for all $ g\neq 0,1,2,3,4,5,7,$ and $ 13$). Then there is an explicit $ N\in \mathbb{Z}$ and an explicit monic polynomial $ f_0(x)\in \mathbb{Z}[x]$ of degree $ n$, such that the Jacobian $ J$ of every curve of the form $ y^2=f(x)$ has $ \operatorname {Gal}(\mathbb{Q}(J[\ell ])/\mathbb{Q})\cong \operatorname {GSp}_{2g}(\mathbb{F}_\ell )$ for all odd primes $ \ell $ and $ \operatorname {Gal}(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, whenever $ f(x)\in \mathbb{Z}[x]$ is monic with $ f(x)\equiv f_0(x) \bmod {N}$ and with no roots of multiplicity greater than $ 2$ in $ \overline {\mathbb{F}}_p$ for any $ p\nmid N$.

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Samuele Anni
Affiliation: Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, Case 907, 163, avenue de Luminy, F13288 Marseille cedex 9, France

Vladimir Dokchitser
Affiliation: Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom

Keywords: Galois representations, abelian varieties, hyperelliptic curves, inverse Galois problem, Goldbach's conjecture
Received by editor(s): June 4, 2019
Received by editor(s) in revised form: August 18, 2019
Published electronically: November 5, 2019
Additional Notes: The first author was supported by EPSRC Programme Grant ‘LMF: L-Functions and Modular Forms’ EP/K034383/1 during his position at the University of Warwick, and by DFG Priority Program SPP 1489 and the Luxembourg FNR during his positions at IWR, Heidelberg and at the University of Luxembourg
The second author was supported by a Royal Society University Research Fellowship
Article copyright: © Copyright 2019 American Mathematical Society