What is Arithmetic Geometry?

Arithmetic geometry is where number theory meets algebraic geometry. Its origins can be traced back to the ancient Greek mathematician Diophantus, who was interested in rational and integral solutions to polynomial equations. Perhaps the most well-known problem of this nature is Fermat's Last Theorem, which asserts that

xⁿ + yⁿ = zⁿ

has no solutions in the integers for n ≥ 3 other than those where x, y, or z is zero. In 1637, Fermat notoriously claimed to have a proof of this assertion but did not write it down, stating instead, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Mathematicians generally believe it was unlikely that Fermat had a complete proof. The problem remained unsolved for over 300 years, until the 1990s, when Andrew Wiles announced a proof that established a profound connection between modular forms and elliptic curves via Galois representations.

In the 21st century, arithmetic geometry has continued to evolve rapidly, greatly benefiting from advances in computing technology. In this area, we often liken computers to “telescopes” as they enable us to see further than ever before and gather tremendous amounts of data. My research in arithmetic geometry aims to advance both theoretical knowledge and computational tools related to the arithmetic of abelian varieties and their Galois representations.

Preprints and Publications

• Rational Solids with Equal Surface Area and Volume [preprint] [code]

  • Joint with Kate Finnerty

  • Preprint, 2026

• On the Density of Coprime Reductions of Elliptic Curves [preprint] [code]

  • Joint with Asimina S. Hamakiotes, Sung Min Lee, and Tian Wang

  • Submitted, 2026

• Rational Points on Modular Curves via Maps to Elliptic Curves with Rank Zero [preprint] [code]

  • Joint with Jeremy Rouse

  • Awarded the Selfridge Prize

  • Accepted in Res. Number Theory (Proceedings of ANTS XVII)

• The Possible Adelic Indices for Elliptic Curves Admitting a Rational Cyclic Isogeny [preprint] [code]

  • Joint with Kate Finnerty, Tyler Genao, and Rakvi

  • Submitted, 2026

• A Uniform Bound on the Smallest Surjective Prime of an Elliptic Curve [journal] [preprint] [code]

  • Joint with Tyler Genao and Jeremy Rouse

  • Ramanujan J. 70 (2026), Art. 22

• Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves [journal] [preprint] [code]

  • Joint with Sung Min Lee and Tian Wang

  • Canad. J. Math., published online 24 June 2025, 1--51

• An Effective Open Image Theorem for Products of Principally Polarized Abelian Varieties [journal] [preprint] [code]

  • Joint with Tian Wang

  • J. Number Theory 274 (2025), 140--179

• On the Effective Version of Serre’s Open Image Theorem [journal] [preprint]

  • Joint with Tian Wang

  • Bull. Lond. Math. Soc. 56 (2024), no. 4, 1399--1416

• Serre Curves Relative to Obstructions Modulo 2 [proceedings] [preprint] [code]

  • Joint with Rakvi

  • LuCaNT: LMFDB, Computation, and Number Theory, Contemp. Math. 796 (2024), 103--128

• Computing Nonsurjective Primes Associated to Galois Representations of Genus 2 Curves [proceedings] [preprint] [code]

  • Joint with Barinder Singh Banwait, Armand Brumer, Hyun Jong Kim, Zev Klagsbrun, Padmavathi Srinivasan, and Isabel Vogt

  • LuCaNT: LMFDB, Computation, and Number Theory, Contemp. Math. 796 (2024), 129--163

• Rigidity in Elliptic Curve Local-Global Principles [journal] [preprint]

  • Acta Arith. 211 (2023), no. 3, 265--288

• A Bound for the Image Conductor of a Principally Polarized Abelian Variety with Open Galois Image [journal] [preprint]

  • Proc. Amer. Math. Soc. Ser. B 9 (2022), 272--285

• Square Curious Numbers [journal] [preprint]

  • Joint with Neelima Borade

  • J. Integer Seq. 24 (2021), no. 7, Art. 21.7.4, 14 pp.

• The Asymptotic Distribution of a Hybrid Arithmetic Function [journal]

  • Joint with Sarah Manski and Nathaniel Zbacnik

  • Integers 15 (2015), Paper No. A28, 16 pp.

Software Repositories

• DavenportConstantDB [code]

  • Description: Database of Davenport constants for abelian groups of order up to 500.

  • Joint with Jessica Han

• ModCrvToEC [code]

  • Description: Magma code for computing maps from a modular curve to an elliptic curve.

  • Joint with Jeremy Rouse

• PointSearchNumberFields [code]

  • Description: Magma code for searching for points on projective curves over number fields.

  • Joint with Jeremy Rouse and Flora Yi