Doctor of Philosophy (Ph.D.)

Mathematics
Dartmouth College
2019

Master of Arts (A.M.)

Mathematics
Dartmouth College
2015

Bachelor of Arts (B.A.)

Mathematics and German
Colgate University
2014

Algebraic Number Theory

Computational Methods

Learning Euler Factors of Elliptic Curves

Learning Euler Factors of Elliptic Curves

We apply transformer models and feedforward neural networks to predict Frobenius traces ap from elliptic curves given other traces aq. We train further models to predict apmod2 from aqmod2, and cross-analysis such as apmod2 from aq. Our experiments reveal that these models achieve high accuracy, even in the absence of explicit number-theoretic tools like functional equations of L-functions. We also present partial interpretability findings.

The moduli space of representations of the modular group into G2

The moduli space of representations of the modular group into G₂

In this paper we construct a large four-dimensional family of representations of the modular group into G2. Precisely, this family is an etale cover of degree 96 of an open subset of the moduli space of such representations. This moduli space has two main components, of dimensions one and four. The one-dimensional component consists of well-studied rigid representations, in the sense of Katz. We focus on the four-dimensional component which consists of representations that are not rigid. We also provide algebraic conditions to ensure that the specializations surject onto G2(Fp) for primes p≥5. These representations give new examples of ϕ-congruence subgroups of the modular group as introduced in previous work.

Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves

Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves

We train machine learning models to predict the order of the Shafarevich-Tate group of an elliptic curve over ℚ. Building on earlier work of He, Lee, and Oliver, we show that a feed-forward neural network classifier trained on subsets of the invariants arising in the Birch--Swinnerton-Dyer conjectural formula yields higher accuracies (>0.9) than any model previously studied. In addition, we develop a regression model that may be used to predict orders of this group not seen during training and apply this to the elliptic curve of rank 29 recently discovered by Elkies and Klagsbrun. Finally we conduct some exploratory data analyses and visualizations on our dataset. We use the elliptic curve dataset from the L-functions and modular forms database (LMFDB).

Supercongruences arising from Ramanujan-Sato Series

Supercongruences arising from Ramanujan-Sato Series 

Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan- Sato series for 1/π, and used this to construct 11 explicit examples of Ramanujan-Sato series aris- ing from modular forms for arithmetic triangle groups of non-compact type. Here, we use work of Chisholm, Deines, Long, Nebe and the third author to prove a general p-adic supercongruence theorem through an explicit connection to CM hypergeometric elliptic curves that provides p-adic analogues of these Ramanujan-Sato series examples.

Zeta Functions for Table Algebras and Fusion Rings with Irrational-Valued Characters

Zeta Functions for Table Algebras and Fusion Rings with Irrational-Valued Characters

We calculate ideal zeta functions for certain orders of rank 3 defined by stan- dard integral table algebras and integral fusion rings that have irrational-valued irreducible characters. The calculations are obtained from explicit calculations of zeta integrals.