To refer to this page use:
|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 count- ing 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.|
|Electronic Publication Date:||1-Jan-2015|
|Citation:||Bhargava, Manjul, Shankar, Arul. (2015). Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. ANNALS OF MATHEMATICS, 181 (191 - 242. doi:10.4007/annals.2015.181.1.3|
|Pages:||191 - 242|
|Type of Material:||Journal Article|
|Journal/Proceeding Title:||ANNALS OF MATHEMATICS|
Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.