Average rank of elliptic curves

Details Average rank of elliptic curves Average rank of elliptic curves

2012-03-05

arxiv.org/abs/1203.0809v1

Mathematics

Number Theory

17 pages; expository article on the work of Manjul Bhargava and Arul Shankar

Scientific paper

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.

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