Counting elliptic curves of bounded Faltings height

Abstract

Because many invariants and properties of elliptic curves are difficult to understand directly, the study of arithmetic statistics instead looks at what happens "on average", using heights to make this notion rigorous. Previous work has primarily used the naive height, which can be calculated easily but is not defined intrinsically. We give an asymptotic formula for the number of elliptic curves over Q with bounded Faltings height. Silverman [34] has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of the minimal discriminant and period of the elliptic curve. We use this to recast the problem as one of counting lattice points in an unbounded region in R2 defined by transcendental equations, and understand this region well enough to give a formula for the number of these lattice points.