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

Manjul Bhargava, Arul Shankar

Research output: Contribution to journalArticle

30 Scopus citations

Abstract

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, theclassical 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 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.

Original languageEnglish (US)
Pages (from-to)191-242
Number of pages52
JournalAnnals of Mathematics
Volume181
Issue number1
DOIs
StatePublished - Jan 1 2015

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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