Research

I am interested in Algebraic Number Theory and Arithmetic Geometry, and am excited to continue to pursue research in these fields (and related areas) in the graduate program. My current research, which is an extension of the work I did for my senior thesis (with advisor Prof. Barry Mazur), concerns finding nontorsion rational points on specific families of elliptic curves. The two known methods to find nontorsion rational points on infinitely many elliptic curves in a given family, parametrized by a variable $t$, are to construct a point via rational functions, that is, to construct a point on an elliptic curve over $\mathrm{<b>\mathbb{Q}</b>}(t)$, and to construct rational points as sums of Galois conjugates of Heegner points on the elliptic curves in the family. The Heegner point method has significantly more wide-reaching applications than the polynomial method, and was used by Gross and Zagier in 1986 to resolve the Birch and Swinnerton-Dyer conjecture under the assumption that the analytic rank of the elliptic curve is 1. I set out to determine if this method can be applied to the family of elliptic curves with $j$-invariant 0, that is the elliptic curves with equation $$y^2=x^3+D\mathrm{.}$$ Under certain congruence conditions on $D$, and assuming that $D$ is a ratio of coprime squarefree integers and $\mathrm{<b>\mathbb{Q}</b>}(\sqrt[3]{D})$ has odd class number, I was able to construct a nontrivial rational point as a sum of Galois conjugates of an appropriate Heegner point. I presented an earlier version of this result at the conference “Mordell’s Conjecture 100 Years Later” at MIT. I posted it as a preprint on arXiv, and submitted it to a journal.

Preprints

Heegner point constructions and fundamental units in cubic fields – arXiv: 2407.12834 (2024), submitted

Senior Thesis

Explicit Modular Parametrizations and Heegner Point Constructions – Harvard Senior Thesis (2023)

Expository Papers

The Average Size of Selmer Groups of Elliptic Curves – for Math 280Y: Arithmetic Statistics (2023)
The Geometric Sieve and Asymptotics for Counting Number Fields – for Math 273X: The Distribution of Class Groups of Global Fields (2023)
Applications of the theta function to sums of squares – for Math 213A: Complex Analysis (2019)
Bounding the rank of rational elliptic curves – for Math 223A: Algebraic Number Theory (2018)
Hawking Radiation and its Implications – for Physics 260: General Relativity (at UC Davis) (2016)

Talks

Heegner Points on $y^2=x^3+p$ – Mordell's Conjecture 100 Years Later, MIT, Cambridge, MA (2024)
The Galois group of the 27 lines on a rational cubic surface – MIT Primes Conference, MIT, Cambridge, MA (2018)