Polynomials, critical values, Hurwitz spaces
My current focus lies in understanding the arithmetic of critical values of polynomials—that is, of their branching points.
This problem, at least in characteristic $0$, can be naturally phrased in terms of rational points on certain Hurwitz moduli spaces, specifically covers of the moduli space of hyperelliptic curves with genus depending on the degree of the polynomials.
For low degrees, these covers turn out to be related to moduli of certain division structures on hyperelliptic Jacobians. I am currently working on explaining—and possibly extending—this relationship, as well as using it to answer explicit questions, like the Equicriticality problem over number fields and other arithmetic fields $K$:
I am also interested in other arithmetic applications of Hurwitz theory, especially in the spirit of the inverse Galois problem.
Arithmetic of (hyper)elliptic curves
My first steps as a researcher in Number Theory were in the area of elliptic curves, specifically in counting rational points.
I have since then been interested in the distribution of ranks, both algebraic and analytic, especially in the case of quadratic twist families. My Master's thesis gives an overview of known results and techniques (up to, but not including, Smith's work) for the congruent number family.
While the main questions about the distribution of ranks seem currently out of reach, there is hope for progress on easier ones. In my thesis I showed, under GRH, the existence of infinitely many congruent number twists (later extended to any quadratic twist family—work to appear soon) with analytic rank $2,4$ or $6$. The algebraic analogue, with the stronger condition of rank exactly $2$, has recently been shown (unconditionally) by Zwyna.
At the moment I am exploring a relationship, arising from the aforementioned study of critical values, between elliptic curves with $3$-torsion, genus $2$ hyperelliptic curves with $(4,4)-$split Jacobian, and the arithmetic dynamics of quartic polynomials.