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Research

Bianchi period polynomials: Hecke action and congruences

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.



Mathematics of Bitcoin: The ECDSA (University of Warwick, 2018)

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.