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

## Author(s): Bhargava, Manjul; Shankar, Arul

To refer to this page use: http://arks.princeton.edu/ark:/88435/pr17m0z
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dc.contributor.authorBhargava, Manjul-
dc.contributor.authorShankar, Arul-
dc.date.accessioned2017-11-21T19:08:51Z-
dc.date.available2017-11-21T19:08:51Z-
dc.date.issued2015-01en_US
dc.identifier.citationBhargava, Manjul, Shankar, Arul. (2015). Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. ANNALS OF MATHEMATICS, 181 (191 - 242. doi:10.4007/annals.2015.181.1.3en_US
dc.identifier.issn0003-486X-
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/pr17m0z-
dc.description.abstractWe 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 count- ing 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.en_US
dc.format.extent191 - 242en_US
dc.language.isoenen_US
dc.relation.ispartofANNALS OF MATHEMATICSen_US
dc.rightsAuthor's manuscripten_US
dc.titleBinary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curvesen_US
dc.typeJournal Articleen_US
dc.identifier.doidoi:10.4007/annals.2015.181.1.3-
dc.date.eissued2015-01-01en_US
dc.identifier.eissn1939-8980-
pu.type.symplectichttp://www.symplectic.co.uk/publications/atom-terms/1.0/journal-articleen_US

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