CJ Dowd's Website

Arborescences of covering graphs, with Sunita Chepuri, Andrew Hardt, Gregory Michel, Sylvester W. Zhang, and Valerie Zhang, in Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 319-346. Based on our work from the 2019 UMN REU. Also presented at the University of Illinois Chicago Undergraduate Mathematics Symposium, November 2, 2019; here is our poster.

We prove a formula describing a combinatorial relationship between the set of spanning trees of a graph vs. the set of spanning trees of a covering graph.

On parameterizations of cyclic N-isogenies and strict K-curves lying above rational points of Y_0^+(N). My undergraduate thesis, submitted in 2021. Unfortunately I later discovered that my supposedly novel result had been done before in a paper by Josep González (Isogenies of polyquadratic Q-curves to their Galois conjugates, see especially Proposition 3.1).

I give an exposition of Ligozat's formulas for Hauptmoduln of rational modular curves in terms of eta products and use them to tabulate rational expressions for the coefficients of the corresponding elliptic curves. Additionally, I give a Diophantine condition that describes when the isogeny between two elliptic K-curves can be defined over the base field.

Virtual resolutions of general points in smooth Fano toric varieties, with Sean McNally from the 2019 UMN REU.

We prove the existence of virtual resolutions of length dim X for general points in several smooth Fano toric varieties.

• Introduction: Matsushima's formula (9/9/24, following Alex Youcis' notes )
• Algebraic de Rham cohomology, Gauss-Manin connection, and the Kodaira-Spencer isomorphism (9/23/24)
• Parabolic cohomology on modular curves (10/14/24, Notes)
• Langlands-Kottwitz method and trace formulae (11/04/24, Notes)
• Constructing Galois representations associated to modular forms of weight 2 (12/02/24 and 12/09/24, Notes)

• Introduction to Brauer Groups (9/1/23)

• Absolute Hodge Cycles (10/17/23)