Author ORCID Identifier

https://orcid.org/0000-0002-2564-5767

Date of Award

2023

Document Type

Thesis (Ph.D.)

Department or Program

Mathematics

First Advisor

John Voight

Abstract

This manuscript consists of two parts. In the first part, we study generalizations of modular curves: triangular modular curves. These curves have played an important role in recent developments in number theory, particularly concerning hypergeometric abelian varieties and approaches to solving generalized Fermat equations. We provide a new result that shows that there are only finitely many Borel-type triangular modular curves of any fixed genus, and we present an algorithm to list all such curves of a given genus.

In the second part of the manuscript, we explore the problem of computing the set of rational points on a smooth, projective, geometrically irreducible curve of genus g>1 over Q. We study the geometric quadratic Chabauty method, which is an effective method for producing a finite set of p-adic points containing the rational points of the curve. This method is due Edixhoven and Lido. We overview the method and discuss explicit algorithms for finding rational points. We also present a comparison is with the classical (cohomological) quadratic Chabauty method.

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