Research
My research interests lie in number theory and arithmetic geometry. In particular, I am interested in both theoretical and computational aspects of the arithmetic of elliptic curves and higher-dimensional abelian varieties, modular and Shimura curves, Galois representations, and low-degree points on varieties.
I am also interested in involving students in research. In the 2024-2025 academic year, I mentored UIC undergrad Raghav Bhutani’s project titled “Replacement dynamics of binary quadratic forms,” supported by the LASURI program.
Preprints
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Point counts, automorphisms, and gonalities of Shimura curves (preprint)
with Pietro Mercuri, Oana Padurariu, and Claudio Stirpe -
Replacement dynamics of binary quadratic forms (preprint)
with Raghav Bhutani -
Counting points on Atkin–Lehner quotients of some genus \(0\) Shimura curves (preprint)
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
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Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level (computations), (arXiv), (pub)
with Oana Padurariu
Research in Number Theory 11, 2 (2025) -
CM points on Shimura curves via QM-equivariant isogeny volcanoes (computations), (arXiv), (pub)
Pacific Journal of Mathematics, Vol. 332 (2024), no. 2, 321–384 -
The least degree of a CM point on a modular curve (computations), (pdf), (pub)
with Pete L. Clark, Tyler Genao, and Paul Pollack
Journal of the London Mathematical Society (2) 105 (2022), no. 2, 825–883
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Chevalley-Warning at the boundary (pdf), (pub)
with Pete L. Clark and Tyler Genao
Expositiones Mathematicae 39 (2021), no. 4, 604–623 -
Classically integral quadratic forms excepting at most two values (computations), (arXiv), (pub)
with Madeleine Barowsky, Will Damron, Andres Mejia, Nolan Schock, and Kate Thompson
Proceedings of the AMS 146 (2018), no. 9, 3661–3677
Conferences and Workshops
Invited Talks
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Genus one Atkin–Lehner quotients of Shimura curves
AMS Sectional Special Session on Recent Advances in the Mathematics of Cryptography – October 2025 -
A volcanic approach to torsion of CM elliptic curves
Oberseminar, Max Planck Institute for Mathematics – August 2025 -
Effective surjectivity of Galois representations of products of elliptic curves over function fields (slides)
AMS Sectional Special Session on Counting and Asymptotics in Number Theory, Hartford, CT – April 2025
Number Theory Seminar, The Ohio State University – March 2025
Number Theory Seminar, Max Planck Institute for Mathematics – August 2025 -
Shimura curves, gonalities, and low-degree points JMM AMS Special Session on Mathematics Informed by Computing, Seattle, WA – January 2025
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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 AMS 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
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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
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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 – November 2019 -
Midwest Arithmetic Geometry and Number Theory Series (MAGNTS)
The Ohio State University – October 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 and Internal Seminar Talks
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Effective surjectivity of Galois representations of products of elliptic curves over function fields
Number Theory Seminar – UIC – February 2025 -
Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level
Number Theory Seminar – UIC – October 2024
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
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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 – November 2019 -
Introduction to Sage and Magma
Street Math and Related Topics Seminar (SMARTS), UGA – November 2019 -
Modular forms part 2
Arithmetic Moduli of Elliptic Curves Reading Seminar, UGA – August 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