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

Manjul Bhargava, Arul Shankar

Research output: Contribution to journalArticle

14 Scopus citations

Abstract

We prove an asymptotic formula for the number of SL3(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.

Original languageEnglish (US)
Pages (from-to)587-621
Number of pages35
JournalAnnals of Mathematics
Volume181
Issue number2
DOIs
StatePublished - Jan 1 2015

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint Dive into the research topics of 'Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0'. Together they form a unique fingerprint.

  • Cite this