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

Manjul Bhargava¹ ; Arul Shankar²

¹ Department of Mathematics, Princeton University, Fine Hall-Washington Rd., Princeton, NJ 08544
² Department of Mathematics, One Oxford Street, Harvard University, Cambridge, MA 02138

Annals of mathematics, Tome 181 (2015) no. 2, pp. 587-621.

Résumé

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.

Détail

DOI : 10.4007/annals.2015.181.2.4