## Columbia Mathematics Department
Colloquium

*The average rank of elliptic curves *

### by

### Manjul Bhargava

### Princeton
University

*Abstract:
*

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**