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

Annals of Mathematics

Description:

Annals of Mathematics, a distinguished journal of research papers in pure mathematics, was founded in 1884. Annals of Mathematics is published bimonthly with the cooperation of Princeton University and the Institute for Advanced Study. It is widely regarded as one of the main mathematics journals in the world.

Coverage: 1884-2020 (Vol. 1, No. 1 - Vol. 191, No. 1)

Moving Wall: 5 years (What is the moving wall?)

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ISSN: 0003486X

EISSN: 19398980

Subjects: Mathematics, Science & Mathematics

Collections: Arts & Sciences I Collection, JSTOR Archival Journal & Primary Source Collection, JSTOR Essential Collection, Mathematics & Statistics Collection

Abstract

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 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.

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

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Abstract