Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

TL;DR

This paper extends classical counting results to binary quartic forms with bounded invariants and uses these results to show that the average rank of elliptic curves over rationals, ordered by height, is bounded, with a mean 2-Selmer group size of 3.

Contribution

It provides a new asymptotic count for binary quartic forms with bounded invariants and applies this to establish bounds on the average rank of elliptic curves.

Findings

Average rank of elliptic curves is bounded when ordered by height.

Mean size of the 2-Selmer group of elliptic curves is 3.

Limsup of the average rank is at most 1.5.

Abstract

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general, and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer group is 3. This implies that the limsup of the average rank of elliptic curves is at most 1.5.