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

By Manjul Bhargava, A. Shankar. 2010

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We prove an asymptotic formula for the number of ${\rm SL}_3({\mathbb Z})$-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate-Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have non-vanishing $L$-function at $s=1$. It follows that a positive proportion of all elliptic curves satisfy BSD.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4ba813b194fe61fece8142ea2cb8f30511b80fcb

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Citations in the corpus

Most hyperelliptic curves over Q have no rational points (2013, methodology): “c(C). In particular, this would mean that the 2-Selmer groups of 100% of these locally soluble curves Chave size exactly 2. To show that this leads to a contradiction, we use the following trick from [10]. Namely, we ﬁrst prove: Theorem 20 A positive proportionof locally soluble genus one curves C: ”
Geometric representations of graded and rational Cherednik algebras (2014, methodology): “The action of PGL(V ) on Sym3(V ∨) ⊗ det(V ) is used by Bhargava and Shanker to calculate the average size of the 3-Selmer groups of elliptic curves over Q, see [6] and”
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, background): “most one rational point, and a majority of such curves have at most 7 rational points; see [40] for details and further such results. Note that Corollary 4(a) is also true when n = 1: it follows from [5] that a positive proportion of elliptic curves have only one rational point (the origin of the gr”