Geodesic
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
Manjul Bhargava1 ; Arul Shankar2
1 Department of Mathematics, Princeton University, Fine Hall - Washington Rd., Princeton, NJ 08544
2 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138
Annals of mathematics, Tome 181 (2015) no. 1, pp. 191-242.

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 $1.5$.
DOI : 10.4007/annals.2015.181.1.3

Manjul Bhargava 1 ; Arul Shankar 2

1 Department of Mathematics, Princeton University, Fine Hall - Washington Rd., Princeton, NJ 08544
2 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138 
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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. http://geodesic.mathdoc.fr/articles/10.4007/annals.2015.181.1.3/

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