Counting elliptic curves of bounded Faltings height

TL;DR

This paper derives an asymptotic formula for counting elliptic curves over the rationals with bounded Faltings height by translating the problem into counting lattice points in a specific region of the plane.

Contribution

It provides a novel asymptotic counting method for elliptic curves based on Faltings height, utilizing modular functions and lattice point enumeration.

Findings

Asymptotic formula for the number of elliptic curves with bounded Faltings height

Reformulation of the counting problem as lattice point enumeration in D

Connection between Faltings height and modular functions

Abstract

We give an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$ with bounded Faltings height. Silverman has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in $\mathbb{R}^2$.