Wild Galois representations of elliptic and hyperelliptic curves

Abstract

In this thesis we investigate the Galois representation attached to an algebraic curve over a local field. In particular, we consider elliptic and hyperelliptic curves with very bad reduction, i.e. which acquire good reduction over a wildly ramified extension.
It is a well-known fact that the primes at which an elliptic curve may acquire good reduction over a wildly ramified extension are 2 and 3. For such primes, a classification of the restriction to inertia of the Galois representation is well understood in the literature; in particular the image of inertia is a finite group, which can be either cyclic or non-abelian. However, less is known about the full Galois action. In this work we give an explicit and algorithmic description of the Galois representations which occur in these cases, with a particular focus on the curves with non-abelian inertia image.
For a hyperelliptic curve of genus g, the primes of wild reduction are at most 2g +1. In this work we consider the family of hyperelliptic curves which have potentially good reduction at 2g +1, assuming this is a prime, and the largest possible image of inertia. We will see how the result on the Galois representation attached to one such curve is a direct generalisation of the
corresponding result for elliptic curves.

Date of Award	23 Mar 2021
Original language	English
Awarding Institution	University of Bristol
Supervisor	Nina C Snaith (Supervisor) & Tim Dokchitser (Supervisor)