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

2010/06/04 by Manjul Bhargava, Arul Shankar, Bhargava, Manjul +1 · 18 citations
Mathematics · #11E76 #11G05 #11R45 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1006.1002

openalex publication_date 2010/06/04 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We prove a theorem giving the asymptotic number of binary quartic forms\nhaving bounded invariants; this extends, to the quartic case, the classical\nresults of Gauss and Davenport in the quadratic and cubic cases, respectively.\nOur techniques are quite general, and may be applied to counting integral\norbits in other representations of algebraic groups.\n We use these counting results to prove that the average rank of elliptic\ncurves over \ℚ, when ordered by their heights, is bounded. In\nparticular, we show that when elliptic curves are ordered by height, the mean\nsize of the 2-Selmer group is 3. This implies that the limsup of the average\nrank of elliptic curves is at most 1.5.\n

Citations

• Arithmetic Subgroups of Algebraic Groups
• On the Davenport-Heilbronn theorems and second order terms

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• Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
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• The second moment of the size of the 2-class group of monogenized cubic fields
• Geometric representations of graded and rational Cherednik algebras
• Counting S₅-fields with a power saving error term
• Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0
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• A Topological Perspective on the Birch and Swinnerton Dyer Conjectures
• Arul Shankar [wikipedia]
• Rank of an elliptic curve [wikipedia]

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