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Title: Aspects of p-adic computation
Author: Doris, Christopher
ISNI:       0000 0004 7968 3034
Awarding Body: University of Bristol
Current Institution: University of Bristol
Date of Award: 2019
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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.
Supervisor: Dokchitser, Tim Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: mathematics ; number theory ; p-adic ; local fields ; ramification ; computation