Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0

TL;DR

This paper develops an asymptotic count of ternary cubic forms with bounded invariants and uses it to show that a positive proportion of elliptic curves have rank 0 or 1, supporting the Birch and Swinnerton-Dyer conjecture.

Contribution

It provides a new counting method for ternary cubic forms and applies it to establish positive proportions of elliptic curves with specific ranks.

Findings

Average size of 3-Selmer group is 4.

Average rank of elliptic curves is less than 1.17.

Positive proportion of elliptic curves have rank 0 or 1.

Abstract

We prove an asymptotic formula for the number of ${\rm SL}_3({\mathbb Z})$-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate-Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have non-vanishing $L$-function at $s=1$. It follows that a positive proportion of all elliptic curves satisfy BSD.