Voir la notice de l'article provenant de la source Annals of Mathematics website
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 $\Bbb{Q}$, 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{10_4007_annals_2015_181_1_3,
author = {Manjul Bhargava and Arul Shankar},
title = {Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves},
journal = {Annals of mathematics},
pages = {191--242},
publisher = {mathdoc},
volume = {181},
number = {1},
year = {2015},
doi = {10.4007/annals.2015.181.1.3},
mrnumber = {3272925},
zbl = {1307.11071},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.181.1.3/}
}
TY - JOUR AU - Manjul Bhargava AU - Arul Shankar TI - Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves JO - Annals of mathematics PY - 2015 SP - 191 EP - 242 VL - 181 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.181.1.3/ DO - 10.4007/annals.2015.181.1.3 LA - en ID - 10_4007_annals_2015_181_1_3 ER -
%0 Journal Article %A Manjul Bhargava %A Arul Shankar %T Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves %J Annals of mathematics %D 2015 %P 191-242 %V 181 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.181.1.3/ %R 10.4007/annals.2015.181.1.3 %G en %F 10_4007_annals_2015_181_1_3
Manjul Bhargava; Arul Shankar. Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Annals of mathematics, Tome 181 (2015) no. 1, pp. 191-242. doi: 10.4007/annals.2015.181.1.3
Cité par Sources :