Jack Thorne

Arithmetic statistics for odd genus 2 curves

An odd genus 2 curve is a hyperelliptic curve of the form y^2 = f(x), where f(x) is a monic degree 5 polynomial. The Jacobian J_f of such a curve is an abelian surface, and if f has rational coefficients then it is of interest to study the Mordell--Weil group J_f(Q), or approximations to this such as the p-Selmer group for primes p.

Bhargava and Gross have studied the average size of the 2-Selmer group of J_f (as f varies). I will discuss joint work with Beth Romano where we do the same for the 3-Selmer group, using a relation with the exceptional Lie group E_8. One consequence of our work (combined with earlier work by Poonen and Stoll) is that a positive proportion of curves C_f have no rational points apart from the marked point at infinity.