Aspects of p-adic computation

Abstract

We present a collection of new algorithms and approaches to several aspects of p-adic computation including:
• computing the Galois group of a polynomial defined over a p-adic field;
• computing the conductor of a 2-adic hyperelliptic curve of genus 2;
• representing p-adic numbers exactly using lazy arithmetic; and
• finding the roots of a system of polynomials in several variables over a p-adic field.

In all cases, these algorithms are new or improve significantly on the previous state of the art. Most are implemented in the Magma computer algebra system, with source code freely available on the author’s website.

We have used these to prove the conductors of all genus 2 curves in the L-functions and modular forms database (LMFDB), which were previously
conjectural, and have verified the Galois groups in the local fields database. We have also produced tables of previously unknown Galois groups, also available on the author’s website.

Date of Award	25 Jun 2019
Original language	English
Awarding Institution	The University of Bristol
Supervisor	Tim Dokchitser (Supervisor)