Columbia Mathematics Department Colloquium


The average rank of elliptic curves


  Manjul Bhargava  

Princeton University



A  rational elliptic curve may be viewed as the set of solutions to an equation of the form y^2=x^3+Ax+B, where

 A and B are rational numbers.  It is known that the rational points on this curve possess a natural abelian group

structure, and the Mordell-Weil theorem states that this group is always finitely generated.  The rank of a rational

elliptic curve measures how many rational points are needed to generate all the rational points on the curve.

There is a standard conjecture, originating in work of Goldfeld, that states that the average rank of all elliptic curves

 should be 1/2; however, it has not previously been known that the average rank is even finite!  In this lecture, we describe

 recent work that shows that the average rank is finite (in fact, we show that the average rank is bounded by 1.5).




  March 24th, Wednesday, 5:00-6:00 pm

Mathematics 520

Tea will be served at 4:30pm