The average rank of elliptic curves over number fields

Abstract

In joint work with Manjul Bhargava, we proved that the average rank of

rational elliptic curves, when ordered by their heights, is bounded above by 1.5. This

result was accomplished by using Bhargava's geometry-of-numbers methods

to obtain asymptotics for the number of GL2 (Z)-orbits on integral

binary quartic forms having bounded invariants.

This thesis extends these methods and generalizes the counting results to the

space of binary quartic forms over the ring of integers of any number field. As a

consequence, we prove that the average rank of elliptic curves over any number field

is at most 1.5.