Research

My research is in the area of arithmetic geometry, which lies in the intersection of number theory and algebraic geometry.  More specifically, I am interested in elliptic and hyperelliptic curves (those are curves defined by an equation of the form y^2 = f(x) where f is a polynomial function) over local and global fields, the Jacobian varieties and Galois actions attached to them, their semistable models, and how they may be uniformized when over local fields.  Lately I've been dipping my toes in superelliptic curves as well (where y^2 can be replaced by y to any power).  My particular angle of focus in recent years has been on describing these things directly from looking at the distances between the roots of the polynomial defining the (hyper/super)elliptic curve, where "distances" are defined in the p-adic sense with respect to a prime of the ground field; this is known as the cluster data of the branch locus.


Below are links to my publications and preprints.

PUBLICATIONS:

10) Clusters, toric ranks, and 2-ranks of hyperelliptic curves in the wild case (joint with Leonardo Fiore, published in Research in Number Theory, May 2026)

9) Boundedness results for 2-adic Galois images associated to hyperelliptic Jacobians
(published in Mathematische Nachrichten, July 2021)

8) Lifting images of standard representations of symplectic groups (published in Manuscripta Mathematica, May 2021)

7) Divisibility of torsion subgroups of abelian surfaces over number fields (coauthored with John Cullinan, published in Canadian Journal of Mathematics, October 2020)

6) Semistable models of elliptic curves over residue characteristic 2 (published in Canadian Mathematical Bulletin, July 2020)

5) Prime-to-p étale fundamantal groups of punctured projective lines over strictly Henselian fields
(coauthored with Hilaf Hasson, published in Transactions of the American Mathematical Society, February 2020)

4) An abelian subextension of the dyadic division field of a hyperelliptic Jacobian (published in Mathematica Slovaca, March 2019)

3) A note on 8-division fields of elliptic curves
(published in European Journal of Mathematics, August 2017)

2) Dyadic Torsion of Elliptic Curves (published in European Journal of Mathematics, July 2015)

1) Images of 2-adic representations associated to hyperelliptic Jacobians
(published in Journal of Number Theory, 2015)



PREPRINTS:

Clusters and semistable models of hyperelliptic curves in the wild case (coauthored with Leonardo Fiore, posted to ArXiv)

Below is another (perhaps more easily digestible) paper covering much of the content of the "clusters" paper listed above:

Clusters and semistable models of hyperelliptic curves in the wild case (15-page announcement version)



A cluster criterion for potential degeneracy of superelliptic curves (posted to ArXiv)

Polynomials whose nth powers have prescribed multiple-of-nth-degree coefficients (posted to ArXiv)

Split degenerate superelliptic curves and ℓ-adic images of inertia (posted to ArXiv)

Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put (posted to ArXiv)

Branch points of split degenerate superelliptic curves I: non-archimedean uniformization (posted to ArXiv and submitted to a journal for publication)



Dyadic Torsion of 2-Dimensional Hyperelliptic Jacobians (posted to ArXiv)


And here is a link to my dissertation (minor edits have been made since officially submitting it):
Hyperelliptic Jacobians and their associated \ell-adic Galois representations


Here are some (slightly sloppier) notes on various background topics which I wrote up just for fun:

Notes on Riemann's Existence Theorem -- proving the algebraic statement of RET from the analytic     statement of RET, mostly following Volklein

Polarization of complex abelian varieties

Symplectic representations of braid groups

Explicit construction of hyperelliptic Jacobians

Semistable models of elliptic curves over residue characteristic 2 -- an early version of the full article of the same title linked to above, except this one contains an extra section about "lifting"