This is the second of two lectures.

A rational elliptic curve is the set of solutions to an equation of the form $y^2=x^3+Ax+B$, where $A$ and $B$ are rational numbers. Elliptic curves have a remarkable property, namely, given two rational points on an elliptic curve, one can draw a line through those points to obtain a third rational point on the curve. In this way, starting with only finitely many points on the curve, one can construct more (even infinitely many!) such points. This generating procedure has had many important applications in number theory, arithmetic geometry, and cryptography.

In this talk, we will describe this elementary construction and how it works. Then, we will address the following question: how many points are needed, on average, to generate all rational points on an elliptic curve?

For more details see http://websites.swarthmore.edu/lchen/Colloquium1011/Bhargava.pdf.