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 integer solutions to polynomial equations with integer coefficients. 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 for 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

  1. A uniform bound on the smallest surjective prime of an elliptic curve [preprint] [code]

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

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

    • Joint with Tian Wang

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

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

    • Joint with Tian Wang

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

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

    • Joint with Rakvi

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

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

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

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

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

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

  9. Square Curious Numbers [journal] [preprint]

    • Joint with Neelima Borade

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

  10. The Asymptotic Distribution of a Hybrid Arithmetic Function [journal]

    • Joint with Sarah Manski and Nate Zbacnick

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

Additional Repositories

1. PointSearchNumberFields [code]