Research

My research interests lie in number theory and arithmetic geometry. In particular, I am interested in (torsion of) elliptic curves and low-dimensional abelian varieties, often in the case of extra endomorphisms and from the vantage of rational points on modular and Shimura curves, and in low-degree points on curves and on higher-dimensional varieties.

I am also interested in involving undergraduates in research. In the Fall 2024 and Spring 2025 semesters, I will be the faculty mentor for UIC undergrad Raghav Bhutani’s project on “replacement dynamics of binary quadratic forms,” supported by the LASURI program.

Preprints

Counting points on Atkin–Lehner quotients of some genus \(0\) Shimura curves (Draft available upon request)
with Tyler Genao, Tristan Phillips, Tim Santens, and John Yin
CM elliptic curves: volcanoes, reality and applications, part II (preprint)
with Pete L. Clark

Publications

CM points on Shimura curves via QM-equivariant isogeny volcanoes (computations), (arXiv)
Pacific Journal of Mathematics, to appear
Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level (computations), (arXiv)
with Oana Padurariu
Research in Number Theory, to appear
The least degree of a CM point on a modular curve (computations), (pdf)
with Pete L. Clark, Tyler Genao, and Paul Pollack
Journal of the London Mathematical Society, 2022

Chevalley-Warning at the boundary (pdf)
with Pete L. Clark and Tyler Genao
Expositiones Mathematicae, 2021
Classically integral quadratic forms excepting at most two values (arXiv)
with Madeleine Barowsky, Will Damron, Andres Mejia, Nolan Schock, and Kate Thompson
Proceedings of the AMS, 2018

Conferences and Workshops

Invited Talks

Low degree points on product surfaces (slides)
MathFest, Indianapolis, IN – August 2024
Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level (slides)
ANTS XVI, MIT – July 2024
Algebra and Number Theory Seminar, Dartmouth College – March 2024
Heegner points on \(X_0(N)\) (slides) (video)
Bhaskaracharya celebrating Harish-Chandra Lecture Series, Virtual – June 2024
A volcanic approach to CM points on Shimura curves (slides)
Algebra & Number Theory Seminar, Emory University – March 2023
JMM Special Session on Arithmetic Geometry Informed by Computation, Boston, MA – January 2023
Classically integral quadratic forms excepting at most two values (slides)
with M. Barowsky, W. Damron, A. Mejia, and N. Schock
Joint Math Meetings, Atlanta, GA – January 2017

Other Conferences and Workshops Attended

The Mordell conjecture 100 years later
MIT – July 2024
Arizona Winter School - Abelian Varieties
University of Arizona – March 2024
Shimura curves in the LMFDB
Dartmouth College – February 2024
Collaborate@ICERM – Explicit Arithmetic of Shimura Curves
ICERM – August 2023
AMS Math Research Community – Explcit Computations with Stacks
Beaver Hollow, New York – June 2023
Integers 2023
UGA – May 2023
Georgia Algebraic Geometry Symposium (GAGS) 2023
UGA – April 2023
Virtual Number Theory and Arithmetic Geometry (VaNTAGe)
Virtual – Fall 2020, Summer 2021, Fall 2021
PCMI Graduate Summer School on
Number Theory Informed by Computation
Virtual – July 2021
Mathematics Teacher-Scholar Symposium (MaTSS)
Virtual – May 2021
Connecticut Number Theory Conference (CTNT)
Virtual – June 2020
Front Range Number Theory Day
Virtual – April 2020
Arithmetic Geometry is Online In Zoom, Everyone (AGONIZE)
Virtual – March 2020
Arizona Winter School - Nonabelian Chabauty
University of Arizona – March 2020
Modular Forms, Arithmetic, and Women in Mathematics (MAAIM)
Emory University – Nov 2019
Midwest Arithmetic Geometry and Number Theory Series (MAGNTS)
Ohio State University – Oct 2019
CMI-HIMR Summer School in Computational Number Theory
University of Bristol – June 2019
Connecticut Summer School in Number Theory and
Conference on Elliptic Curves and Modular Forms (CTNT)
University of Connecticut – May 2018

Contributed Talks

Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level
CTNT – University of Connecticut – June 2024
An ultrametric space oddysey
Math Club – UIC – February 2024
Towards a better understanding of Shimura curves
Number Theory Seminar – UIC – September 2023
A volcanic approach to CM points on Shimura curves
Colloquium – UIC – September 2023
SRNTC, Louisiana State University – March 2023
PANTS XXXV, University of South Carolina – December 2022
Number Theory Seminar, UGA – September 2022
Poonen’s Theorem
Drinfeld modules learning seminar, UGA – April 2023
Explicit class field theory for Drinfeld modules
Drinfeld modules learning seminar, UGA – February 2023
On Hindry–Silverman’s Sur le nombre de points de torsion rationnels sur une courbe elliptique Short and Sweet Seminar (SASS), UGA – October 2022
The Chebotarev density theorem
Catalan’s Conjecture after Mihailescu (CCAM) Reading Seminar, UGA – November 2021
Isogeny volcanoes
Graduate Student Seminar, UGA – October 2021
Fields of moduli, rationality, and fake elliptic curves
Graduate Summer Conference, UGA – July 2021
On Honda’s Isogeny classes of abelian varieties over finite fields
Classical Readings in Arithmetic/Algebraic Geometry Seminar (CRAAG), UGA – July 2021
What is a Shimura curve?
Graduate Student Seminar, UGA – April 2021
Locally algebraic representations
Serre on Zoom (SeZoom) reading seminar, UGA – March 2021

Galois action on CM points
Zannier on Zoom (ZaZoom) reading seminar, UGA – October 2020
No elliptic curve has a Q-rational point of order 13
Graduate Summer Conference, UGA – July 2020
Sporadic CM points on Modular Curves
Graduate Student Seminar, UGA – April 2020
Formalism of moduli problems
Arithmetic Moduli of Elliptic Curves Reading Seminar, UGA – Nov 2019
Introduction to Sage and Magma
Street Math and Related Topics Seminar (SMARTS), UGA – Nov 2019
Modular forms part 2
Arithmetic Moduli of Elliptic Curves Reading Seminar, UGA – Aug 2019
Triangular numbers and modular forms
Graduate Summer Conference, UGA – July 2019
Introduction to modular forms
Graduate Algebraic Geometry Seminar, UGA – March 2019
An axiomatic approach to homology theory
Graduate Summer Conference, UGA – July 2018