LEAPS-MPS: Geometric Arithmetic Statistics: on the Relative Abundance of Points on Plane Curves

Diophantine equations are polynomial equations for which one studies integral or rational solutions. The solution set of a polynomial in three variables in which every monomial has the same degree is referred to as a plane curve. A famous example of such a plane curve is given by the equation relating Pythagorean triples. The question: “what is the set of degrees D such that equations of the form f(x,y,z) = 0 have solutions of degree D?” is very simple to state, but difficult to characterize. It turns out that the difficulty in characterizing solutions to these equations increases greatly in the degree of the equation. The index of a curve, the greatest common divisor of all the degrees D in this set, is notably difficult to determine. Several examples of curves exist where the index is provably known, however, several significant results in this area only hold for positive proportions of curves. This project will explore questions of this nature. In addition, the investigator will organize a two-week workshop on geometric arithmetic statistics for beginning graduate students with a goal of broadening the participation of mathematicians from underrepresented groups in the field. The central goal of this project is to study the index of a plane curve. The investigator aims to characterize certain plane curves each with index strictly greater than 1 and to find bounds on the proportions of such plane curves. The investigator also plans to study the relationship between a combinatorial object related to the equation of a plane curve and the index of the curve. In addition, the investigator proposes to study degrees of points on trigonal superelliptic curves and the failure of the Hasse principle in families of superelliptic curves. The investigator will study such questions in arithmetic statistics related to points on curves using tools from analytic number theory and arithmetic geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.