Research

My primary research interest lies in Arithmetic Geometry, a field of research where geometry and number theory meet. Most of my research questions can be reduced to a following statement;

Let E be an elliptic curve over Q. Let p be a prime of good reduction, and hence E_p is an elliptic curve over F_p. Consider a geometric property P an elliptic curve over a finite field can satisfy. Consider; given E/Q, how many primes are there that make E_p satisfy the property P? If there are infinite, what is the density of such primes?

Preprints

1.  Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves; preprint (Link)

   • Joint work with Jacob Mayle and Tian Wang

2.  On the Average Congruence Class Bias for Cyclicity and Divisibility of the Groups of Fp-points of Elliptic Curves; preprint (Link); Submitted

3.  On the Acyclicity of Elliptic Curves Modulo Primes in Arithmetic Progressions; preprint (Link); Submitted

   • Joint work with Nathan Jones

In Preparation

1.  Keywords: Drinfeld Modules; Cyclic Reduction

Thesis

• On the Distribution in Arithmetic Progressions of Primes of Various Properties Related to Elliptic Curves (Link)