Papers

Details about my publications are also kept on my CV.

Published

Ordinary Abelian Varieties: Isogeny Graphs and Polarizations, Research in Number Theory, to appear, 2026 (Proceedings of ANTS XVII, the Seventeenth Algorithmic Number Theory Symposium) (with E. Costa, S. Marseglia, D. Roe, and C. Vincent) — awarded the Most Reproducible Software prize by MaRDI and the ANTS XVII organizing committee.
Labeling Abelian Varieties over Finite Fields, in LuCaNT: Databases, Algorithms, and Computational Number Theory, Contemporary Mathematics, vol. 840, AMS, 2026, 277–283 (with E. Costa, S. Marseglia, D. Roe, and C. Vincent) — AMS.
A Rigid Analytic Proof that the Abel-Jacobi Map Extends to Compact-Type Models, Canadian Mathematical Bulletin 67(3):648–654, 2024 (with J. Rabinoff).
The Theory of the Entire Algebraic Functions, Int. Math. Res. Not. IMRN 2024, no. 10, 8609–8617, 2024 (with E. Hrushovski).
Angle Ranks of Abelian Varieties, Math. Ann. 389(2024), no.1, 169–185 (with K. Kedlaya and D. Zureick-Brown).
Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB, in Arithmetic Geometry, Number Theory, and Computation, Simons Symposia, Springer, 2021, 375–448 (with K. Kedlaya, D. Roe, and C. Vincent).
Counterexamples to a Conjecture of Ahmadi and Shparlinski, Exp. Math. 32, no.3, 540–544, 2023 (with K. Kedlaya, D. Roe, and C. Vincent).
Deligne-Illusie Classes as Arithmetic Kodaira-Spencer Classes, Journal de Théorie des Nombres de Bordeaux 31, no. 2, 371–383, 2019 (with D. Zureick-Brown).
Total p-Differentials on Schemes Over Z/p², Journal of Algebra 524, 110–123, 2019 (with E. Katz, J. Rabinoff, and D. Zureick-Brown).
Examples of Geometric Bombieri-Lang-Noguchi Outside Mordell-Lang: Nonrigid Varieties with Ample but Not Globally Generated Cotangent Bundle, Journal of Number Theory 175, 158–166, 2017.
Arithmetic Differential Equations on GL_n I: Differential Cocycles, Journal of Algebra 454, 273–291, 2016 (with A. Buium).
Bits of 3^n in Binary, Wieferich Primes and a Conjecture of Erdős, Journal of Number Theory 158, 268–280, 2016 (with D. Weirich).
Arithmetic Differential Equations on GL_n III: Galois Groups, Selecta Mathematica 22(2), 529–552, 2016 (with A. Buium).
Arithmetic Differential Equations on GL_n II: Arithmetic Lie-Cartan Theory, Selecta Mathematica 22(2), 447–528, 2016 (with A. Buium).
Positivity and Lifts of the Frobenius, Math. Res. Lett. 21, no. 2, 289–295, 2014.
Arithmetic Deformation Theory, PhD thesis, The University of New Mexico, 2013.

Preprints

The Dimension Conjecture Implies The Jacobi Bound Conjecture (with D. Zureick-Brown).
The Jacobi Bound Conjecture for Generically Reduced Differential Schemes (with D. Zureick-Brown).
Order One Differential Equations on Nonisotrivial Algebraic Curves (with J. Freitag).
The Basic Theory of Clifford-Bianchi Groups for Hyperbolic n-Space (with A. Hilado, C. Ingalls, A. Logan).
Ford Spheres in the Clifford-Bianchi Setting (with S. Backman, A. Hilado, and V. Potter).

ABC Conjecture Manuscripts

The Statement of Mochizuki's Corollary 3.12: Initial Theta Data, The Ramanujan Journal 68 (2025), no. 1, article 18 (with A. Hilado). ( journal copy )
The Statement of Mochizuki's Corollary 3.12: the First Two Indeterminacies (with A. Hilado), preprint.
Log-Kummer Correspondences and the Third Indeterminacy (with A. Hilado) — appendix of interpretation tables.
Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12 (with A. Hilado).

Software

magma-clifford-algebras — a Magma package for Clifford algebras and Clifford-Bianchi groups.
magma-diff-alg — a Magma package for differential algebra.
mathcity — an orchestration pack for the Gas City system: it routes machine-generated mathematics through review gates before a human sees it.

Course notes

Algebraic Theory of Differential Equations — book-length notes from the course at UVM, Fall 2022.
Algebraic Curves and Surfaces — book-length notes from the course at UVM, Fall 2021.

Miscellaneous notes

See also my YouTube channel for expository and research videos. Some additional notes appear in the talks page.

Automorphisms of the Affine Line over Non-Reduced Rings (not intended for publication).
Lifted Torsors of Lifts of the Frobenius on Algebraic Curves.
Quick introduction to p-jets.
How units are often used to determine mathematical models (as explained to me by Hannes Schenk).
Deninger Cohomology Theories.
Belyi's Theorem (half of it as explained to me by Zach Scherr).
Quadratic Reciprocity — notes from when I subbed for Danielle and didn't finish in class.
Model Theory notes — taken from Thomas Scanlon's course at the CUNY Arithmetic Dynamics Conference, July 2010.
Eichler-Shimura (weight two) — a crash course in modular parametrizations of elliptic curves (the converse of what Wiles proved, the easy part).
History of Jet Spaces and Diophantine Geometry — a section excluded from my research statement.
p-adic and Dyadic Harmonic Analysis — Calderón-Zygmund operators.
Harmonic Analysis, Metric Spaces and PDEs (edited with Lucas Chaffee, Rafael Espinosa, Jarod Hart, Anna Kairema, Lyudmila Korobenko).
Zeros of Analytic Functions That Don't Have Zeros (Szegő Curves) — undergraduate work with Ken McLaughlin at the University of Arizona (VIGRE), with movies.
Interpretations in IUT.