Anni, S;
Dokchitser, V;
(2019)
Constructing hyperelliptic curves with surjective Galois representations.
**Transactions of the American Mathematical Society**
10.1090/tran/7995.
(In press).

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## Abstract

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-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 l 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 appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N 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 $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and $Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, 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 not dividing N.

Type: | Article |
---|---|

Title: | Constructing hyperelliptic curves with surjective Galois representations |

Open access status: | An open access version is available from UCL Discovery |

DOI: | 10.1090/tran/7995 |

Publisher version: | https://doi.org/10.1090/tran/7995 |

Language: | English |

Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |

Keywords: | math.NT, math.NT, 11F80 (Primary), 12F12, 11G10, 11G30 (Secondary) |

UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |

URI: | https://discovery.ucl.ac.uk/id/eprint/10089730 |

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