A Bound on The Average Rank of j-Invariant Zero Elliptic Curves

Abstract

In this thesis, we prove that the average rank of j-invariant 0 elliptic curves, when ordered by discriminant, is bounded above by 3. This work follows from work of Bhargava and Shankar relating elements of the 2-Selmer groups of elliptic curves with equivalence classes of certain binary quartic forms. We also count the number of equivalence classes of these binary quartic forms. This step involves counting the number of points on a quadric in a homogenously expanding non-compact region. To count the number of points on this quadric, we use a modified version of the circle method. This work also has an application to the statistics of the class group of certain pure cubic fields.