Research projects

5. First explicit reciprocity law for unitary Friedberg-Jacquet periods

2023

math.nt

Consider a unitary group $G(\mathbb{A}_{F^+})=U_{2r}(\mathbb{A}_{F^+})$ over a CM extension $F/F^+$ with $G(\mathbb{A}_\infty)$ compact. In this article, we study the Beilinson--Bloch--Kato conjecture for motives associated to irreducible cuspidal automorphic representations $\pi$ of $G(\mathbb{A}_{F^+}).$ We prove that if $\pi$ is distinguished by the unitary Friedberg--Jacquet period, then the Bloch--Kato Selmer group (with coefficients in a favorable field) of the motive of $\Pi=\mathrm{BC}(\pi)$ vanishes.

4. Spherical functions on symmetric spaces of Friedberg-Jacquet type

2023

math.nt

We give explicit models for spherical functions on $p$ -adic symmetric spaces $X=H\backslash G$ for pairs of $p$ -adic groups $(G,H)$ of the form $(\mathrm{U}(2r),\mathrm{U}(r)\times \mathrm{U}(r)),$ $(\mathrm{O}(2r),\mathrm{O}(r)\times \mathrm{O}(r)),$ $(\mathrm{Sp}(4r),\mathrm{Sp}(2r)\times\mathrm{Sp}(2r))),$ $(\mathrm{U}(2r+1),\mathrm{U}(r+1)\times \mathrm{U}(r)),$ and $(\mathrm{O}(2r+1),\mathrm{O}(r+1)\times \mathrm{O}(r)).$ As an application, we completely describe their Hecke module structure.

arXiv →

3. Spherical functions of symmetric forms and a conjecture of Hironaka

2023

math.nt

For all $r\ge1,$ we verify the following conjecture of Hironaka: for a $p$ -adic field $F$ with $p$ odd, the space of spherical functions of $\mathrm{Sym}_{r\times r}(F)\cap\mathrm{GL}_r(F)$ is free of rank $4^r$ over the Hecke algebra.

arXiv →

2. On Howard's main conjecture and the Heegner point Kolyvagin system

2019

math.nt

princeton senior thesis

We upgrade Howard's divisibility toward Perrin-Riou's Heegner point Main Conjecture to an equality under some mild conditions. We do this by exploiting Wei Zhang's proof of the Kolyvagin conjecture. The main ingredient is an improvement of Howard's Kolyvagin system formalism. As another consequence of it, we establish the equivalence between this main conjecture and the primitivity of the Kolyvagin system in certain cases, by also exploiting a explicit reciprocity law for Heegner points.

arXiv →

1. A proof of Kolyvagin's conjecture via the BDP main conjecture

2019

math.nt

princeton junior thesis

We adapt Wei Zhang's proof of Kolyvagin's conjecture for modular abelian varieties over $\mathbb{Q}$ to rely on the BDP main conjecture instead of on the cyclotomic main conjecture. The main ingredient is a reduction to a case that is tractable by the BDP main conjecture, in a similar spirit to Zhang's reduction to the rank one case. By using the BDP main conjecture instead of the cyclotomic main conjecture, our approach is more suitable than Zhang's to extend to modular abelian varieties over totally real fields.

arXiv →

Teaching & Mentorship

Teaching at Johns Hopkins

Fall 2024: 110.617 Number Theory I

High School Enrichment Program Teacher 2021-2024

Virtual classes with students from my former high school on undergraduate-level topics in number theory. notes →

Graduate TA at MIT

Fall 2023: 18.701 Algebra I
Fall 2023: 18.950 Differential Geometry
Fall 2022: 18.02 Multivariable Calculus
Spring 2021: 18.065 Matrix Methods in Data Analysis, Signal Processing & Machine Learning
Fall 2021: 18.701 Algebra I
Fall 2021: 18.700 Linear Algebra
Spring 2020: 18.702 Algebra II
Fall 2020: 18.100A Real Analysis

MIT directed reading program mentor

Winter 2022: Analytic Number Theory
Winter 2020: Modular Forms and Elliptic Curves

Organizing

Fall 2024: Co-organized Number Theory learning seminar on Euler systems

Fall 2022: Learning seminar on Euler systems website →

Talks

Duke

Oct 2024

video →