Binary quartic forms having bounded invariants, and the boundedness of
the average rank of elliptic curves
M. Bhargava, and A. Shankar. (2010)cite arxiv:1006.1002Comment: Proofs have been considerably streamlined, and a number of clarifying details have been added; 36 pages.
Abstract
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 $Q$, 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.
Description
[1006.1002] Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
%0 Journal Article
%1 bhargava2010binary
%A Bhargava, Manjul
%A Shankar, Arul
%D 2010
%K geometry mathematics proof-systems
%T Binary quartic forms having bounded invariants, and the boundedness of
the average rank of elliptic curves
%U http://arxiv.org/abs/1006.1002
%X 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 $Q$, 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.
@article{bhargava2010binary,
abstract = {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.},
added-at = {2020-01-25T21:23:06.000+0100},
author = {Bhargava, Manjul and Shankar, Arul},
biburl = {https://www.bibsonomy.org/bibtex/267cacdcc26f3d1d5c9013b0963047c75/kirk86},
description = {[1006.1002] Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves},
interhash = {d50ff51ef7f1eb89bc3997bc7b3a3f3f},
intrahash = {67cacdcc26f3d1d5c9013b0963047c75},
keywords = {geometry mathematics proof-systems},
note = {cite arxiv:1006.1002Comment: Proofs have been considerably streamlined, and a number of clarifying details have been added; 36 pages},
timestamp = {2020-01-25T21:23:06.000+0100},
title = {Binary quartic forms having bounded invariants, and the boundedness of
the average rank of elliptic curves},
url = {http://arxiv.org/abs/1006.1002},
year = 2010
}