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

By Manjul Bhargava, A. Shankar. 2010

239 citations recorded by the source corpus.

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 $\mathbb{Q}$, 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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6f99f4857cc53066f0099be58c9a833e42dd2999

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Citations in the corpus

Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves (2013, methodology): “To define density of a subset of E precisely, one orders E by height as explained in the introductions to [BS13a] and [PR12].”
Gradings of positive rank on simple Lie algebras (2012, unknown): “2 1⇒ 0 0 0 · · · 0 0 ⇒ 0 −1× [1] μ 2 2(2n+1) 2k+1 , k > 0 1⇒ 0 · · · 0 } {{ } A2k 1 0 · · · 0 } {{ } A2k 1 · · · 1 0 · · · 0 } {{ } A2k ⇒ 1 −1× [d] μ 2”
Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 (2010, methodology): “As in [8], we simplify the counting by averaging over lots of such domains, i.”
Most odd degree hyperelliptic curves have only one rational point (2013, result): “Their work was inspired by the connection between pencils of quadrics and hyperelliptic Jacobians (work of Reid [Rei72], Donagi [Don80], and Wang [Wan12]), and by earlier work by Bhargava and Shankar for elliptic curves [BS10], itself preceded by work of de Jong [dJ02] in the function”
Most hyperelliptic curves over Q have no rational points (2013, methodology): “ over lots of such fundamental regions, i.e., by averaging over the domains FhL(m,τ) where hranges over a certain compact subset G0 ∈ G′(R). The method, which is an extension of the method of [3] and [9], is described next. 15 4.2 Counting irreducible integral points of bounded height In this sectio”
Geometric representations of graded and rational Cherednik algebras (2014, methodology): “Bhargava and Shankar to compute the average size of 2-Selmer groups of elliptic curves over Q, see [5] and [19].”
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, methodology): “nly if the quartic form T(x,y) represents a square in Qv for all places v, and the homogeneous space PT is trivial if and only if T(x,y) has a linear factor over Q. This is the point of view taken in [4], where the latter orbits are called reducible. In order to obtain Theorem 1 from Theorem 5, we a”