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

2010/06/30 by Manjul Bhargava, Arul Shankar, Bhargava, Manjul +1 · 5 citations
Mathematics · #11E76 #11G05 #11R45 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1007.0052

openalex publication_date 2010/06/30 · openalex created_date 2022/08/19 · openalex updated_date 2026/07/28

Abstract

We prove an asymptotic formula for the number of rm SL₃( mathbb\nZ)-equivalence classes of integral ternary cubic forms having bounded\ninvariants. We use this result to show that the average size of the 3-Selmer\ngroup of all elliptic curves, when ordered by height, is 4. This implies that\nthe average rank of all elliptic curves, when ordered by height, is less than\n1.17.\n Combining our counting techniques with a recent result of Dokchitser and\nDokchitser, we prove that a positive proportion of all elliptic curves have\nrank 0. Assuming the finiteness of the Tate-Shafarevich group, we also show\nthat a positive proportion of elliptic curves have rank 1. Finally, combining\nour counting results with the recent work of Skinner and Urban, we show that a\npositive proportion of elliptic curves have analytic rank 0; i.e., a positive\nproportion of elliptic curves have non-vanishing L-function at s=1. It\nfollows that a positive proportion of all elliptic curves satisfy BSD.\n

Cited by

• Unbounded average Selmer ranks of elliptic curves in torsion families
• Selmer groups of families of elliptic curves with an ℓ-isogeny
• Families of curves in Vinberg representations
• Iwasawa Theory of Elliptic Curves in Quadratic Twist Families
• Counting elliptic curves over ℚ with bounded naive height

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