Bjorn Poonen
Published 2012-03-05, updated 2015-06-14Version 2
Bhargava and Shankar prove that as E varies over all elliptic curves over Q, the average rank of the finitely generated abelian group E(Q) is bounded. This result follows from an exact formula for the average size of the 2-Selmer group, which in turn follows from an asymptotic formula for the number of binary quartic forms over Z with bounded invariants. We explain their proof, as well as other arithmetic applications.
Comments: 17 pages. The construction in the second half of Section 4.1 of [BS15b] of a positive-density family of elliptic curves in which the root number is equidistributed is actually taken from p. 25 and Section 9 of Siman Wong's article [Won01]; we have edited our text to credit Wong with this discovery. Also, references have been updated
Journal: S\'eminaire Bourbaki, Vol. 2011/2012, Expos\'es 1043-1058, Ast\'erisque 352 (2013), Exp. No. 1049, 187-204
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
Families of elliptic curves ordered by conductor
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