Research

Creating a dynamic database of finite groups

With John Jones, Jennifer Paulhus, David Roe, Manami Roy and Sam Schiavone

This paper is about a database of finite groups that has been added to the LMFDB. The paper discusses the functional aspects of the database, some algorithmic challenges in its creation and their solutions, and its connectivity with other databases in the LMFDB. As a personal note, since being involved with this project, every time I encounter a new group, I look it up in this database so I can learn the basic facts about it quickly and easily. I highly recommend this approach to studying groups!

Expressions for weight 2 cusp forms in holomorphic eta quotients

With Elisabeth (Yin Ting) Chan

In this paper we report on computations associating expressions in terms of the Dedekind eta function to some cusp forms. In particular, we compute expressions for all the rational weight 2 cusp forms of level up to 100 (with trivial character), except for four, which lie outside the current feasible range of computation. We also devise some notions of minimality of these expressions (as they are often very far from unique), and show how they can be used to verify the locations of zeros of the given form. This paper grew out of computations perform as part of a project for the University of Sheffield's Undergraduate Research Internship scheme.
The code used to compute the expressions is available on my GitHub.

Bianchi period polynomials: Hecke action and congruences

Research in number theory 10, 40 (2024).
arXiv version // Published version

This paper concerns computations with Bianchi modular forms via their period polynomials. In analogy with the classical case, Bianchi modular forms also have a period polynomial made out of L-values, and they are supposed to capture many of the same things classical period polynomials do. In this paper, a method to compute the Hecke action on period polynomials is presented, as well as some methods to (conjecturally) use these polynomials to detect congruences between Bianchi modular forms. We prove congruences between some genuine cuspidal Bianchi forms and
(i) the base-change of an Eisenstein series, and
(ii) the base-change of the discriminant modular form.
The primes appearing in these congruences are 173 and 43 respectively.
The code used to compute the examples in the paper, as well as period polynomials associated to Bianchi modular forms generally, is available on my GitHub.

My Masters thesis investigates the theory underlying elliptic curve encryption, including some background on discrete logarithms, general attacks on elliptic curve encryption, and how these apply to the curve used in verifying Bitcoin transactions, Secp256k1. I think it is broadly free of mathematical errors, although I have not thought about the ECDSA in some time, and corrections are gratefully receieved! It is likely riddled with bad Tex practice.