## Constructing hyperelliptic curves with surjective Galois representations

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- by Samuele Anni and Vladimir Dokchitser PDF
- 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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## Additional Information

**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
**Vladimir Dokchitser**- Affiliation: Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom
- MR Author ID: 768165
- Email: vladimir.dokchitser@kcl.ac.uk
- 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 - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 1477-1500 - MSC (2010): Primary 11F80; Secondary 12F12, 11G10, 11G30
- DOI: https://doi.org/10.1090/tran/7995
- MathSciNet review: 4068270