Local Galois representations for higher genus curves
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
The aim of this research project is to describe the Galois representation attached to a hyperelliptic curve over a local field. Partial results on this problem exist: in particular, for elliptic curves it is known how to compute the inertia image and, in some cases, even the whole Galois representation. The first step of this project is to complete all the remaining cases, giving an explicit solution which can be implemented in a computer program. For higher genus curves, less is known, so the final objective is to understand the Galois representation for special families of curves, for which recent developments in the theory of models of curves can be applied, and again provide an explicit description of it.
People |
ORCID iD |
| Tim Dokchitser (Primary Supervisor) | |
| Nirvana Coppola (Student) |
Publications
Coppola Nirvana
(2018)
Wild Galois Representations: Elliptic curves over a $3$-adic field
in arXiv e-prints
Coppola N
(2020)
Wild Galois representations: elliptic curves over a 3-adic field
in Acta Arithmetica
Coppola N
(2020)
Wild Galois representations: Elliptic curves over a 2-adic field with non-abelian inertia action
in International Journal of Number Theory
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509619/1 | 01/10/2016 | 30/09/2021 | |||
| 1961436 | Studentship | EP/N509619/1 | 01/10/2017 | 31/03/2021 | Nirvana Coppola |
| Description | The main result of the work funded through this award consists of explicitly determine the Galois representation associated to an elliptic or hyperelliptic curve with bad reduction. This problem had already been tackled previously in some special cases, but an explicit general result was not yet available. My contribution consisted of finishing the classification for the elliptic curves case and find some generalisation of this to a special family of higher genus curves. In my PhD thesis I give a complete overview of the cases that were known previously to my research for the Galois representation attached to an elliptic curve, and classify all the remaining cases, which all involve curves with potentially good reduction acquired over a wildly ramified extension. The first case considered is that of an elliptic curve over a 3-adic field which has non-abelian inertia image under the Galois representation. This is an extremal case, in the sense that 3 is the largest prime at which an elliptic curve can have wild reduction and the degree of the minimal extension at which it acquires good reduction (hence the image of inertia) is as large as possible. The result obtained in this case can be proved in more generality for hyperelliptic curves given by an equation of degree p over a p-adic field, which acquire good reduction over a wildly ramified extension which has the largest possible degree. This result is treated in an unpublished (but submitted) paper (see link below) and is included in the final chapter of my thesis. |
| Exploitation Route | An immediate use of these results is an implementation in a computer software in order to compute the Galois representations of a given curve (for elliptic curves it has been done by my supervisor and myself, and for the hyperelliptic case it is work in progress); moreover explicit understanding of Galois representations can be used in other areas of number theory, related to arithmetic invariants associated to curves, such as the computation of conductors and root numbers. In a recent paper by M. Bisatt, the root number for the family of hyperelliptic curves considered in my work is computed explicitly. |
| Sectors | Other |
| URL | https://arxiv.org/abs/2001.08287 |
| Description | Modular method for the Asymptotic Fermat's Last Theorem |
| Organisation | University of Barcelona |
| Country | Spain |
| Sector | Academic/University |
| PI Contribution | The aim of this collaboration is to combine my knowledge of explicit computation of Galois representations attached to elliptic curves and previous work of my collaborator to improve some known results on the (generalised) asymptotic Fermat's Last Theorem. |
| Collaborator Contribution | My partner already has a paper where he attacks some cases of the generalised Fermat's equation to give results on the line of the Asymptotic Fermat's Last Theorem for these equations, via the solution of S-units equations and the modular method. |
| Impact | There is not output yet. |
| Start Year | 2020 |