# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Constructing hyperelliptic curves with surjective Galois representationsHTML articles powered by AMS MathViewer

by Samuele Anni and Vladimir Dokchitser
Trans. Amer. Math. Soc. 373 (2020), 1477-1500 Request permission

## 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
• MR Author ID: 1068732
• Email: samuele.anni@gmail.com