UBC Number Theory Seminar: Shubhrajit Bhattacharya

Arithmetic Statistics is an emerging subbranch of Number Theory where we count arithmetic objects when bounded by some quantitative invariant, like height. One example is counting polynomials with integer coefficients having a fixed Galois group. Bhargava’s proof of B. L. van der Waerden’s conjecture about the count of integer polynomials of degree n and bounded largest coefficient with Galois group not equal to the full symmetry group of order n is a recent breakthrough.

We will introduce a new way of parametrizing monic integer cubic polynomials with Galois group C3 using rational points on a toric variety. We also introduce a new height function on polynomials arising from height functions in toric geometry. This results in a nice relation between the height zeta function of the toric variety, defined by Batyrev–Tschinkel, and the Dirichlet series attached to the counting sequence of monic abelian cubics! Using this we prove explicit and asymptotic formulas for the number of monic abelian cubics of a given height.

We also discuss future research avenues using our method. This is based on joint work with Andrew O’Desky. Preprint available here https://arxiv.org/abs/2310.17831.