Elliptic Curves Ordered by Faltings Height

Abstract

The Faltings height — and the finiteness of semistable principally polarized abelian varietiesof a given genus g over a number field K with bounded Faltings height — is a key ingredient in Faltings’s proof of the Mordell Conjecture. In the case where g = 1 and K = Q, Hortsch gave an expression for the number of elliptic curves with Faltings height less than X. In this thesis, we establish asymptotics for the number of elliptic curves with bounded Faltings height over any number field. We also prove similar asymptotics for a more general class of heights. In addition, we compute an exact expression for the leading term in the asymptotic expression due to Hortsch. One primary motivation for Hortsch’s result is to allow for the computation of arithmetic statistics of elliptic curves with respect to the ordering given by the Faltings height. Bhargava and Shankar computed the average size of the n-Selmer groups of elliptic curves over Q for n ≤ 5 when ordering by naive height. We show that these averages are the same not only for the Faltings height but also for a more general class of heights that includes both the naive height and the Faltings height. These results also hold for any number field.