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

We prove an asymptotic formula for the number of $\mathrm{SL}_3(\mathbb{Z})$-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 equal to 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 nonvanishing $L$-function at $s=1$. It follows that a positive proportion of all elliptic curves satisfy BSD.

Pages 587-621 by Manjul Bhargava, Arul Shankar | From volume 181-2

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

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 $\Bbb{Q}$, 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 $1.5$.

Pages 191-242 by Manjul Bhargava, Arul Shankar | From volume 181-1

The density of discriminants of quintic rings and fields

Pages 1559-1591 by Manjul Bhargava | From volume 172-3

Higher composition laws IV: The parametrization of quintic rings

Pages 53-94 by Manjul Bhargava | From volume 167-1

The density of discriminants of quartic rings and fields

Pages 1031-1063 by Manjul Bhargava | From volume 162-2

Higher composition laws III: The parametrization of quartic rings

Pages 1329-1360 by Manjul Bhargava | From volume 159-3

Higher composition laws II: On cubic analogues of Gauss composition

Pages 865-886 by Manjul Bhargava | From volume 159-2

Higher composition laws I: A new view on Gauss composition, and quadratic generalizations

Pages 217-250 by Manjul Bhargava | From volume 159-1