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 wellknown 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 nonabelian. 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 nonabelian 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 

Supervisor  Nina C Snaith (Supervisor) & Tim Dokchitser (Supervisor) 
Keywords
 Galois representations
 Elliptic curves
 Hyperelliptic curves
 Local fields