The Congruent Number Problem and Elliptic Curves

The congruent number problem is easily stated: given a rational number q, can we find a right triangle with rational sides that has area equal to q?  As it turns out, this problem is intimately related to the problem of finding certain rational points on elliptic curves. In this talk we'll define an elliptic curve, investigate how to transfer the addition law from the complex plane to our curve (thereby turning it into an abelian group), and find all the points of finite order on a certain family of these curves. Using these fancy notions, we'll briefly discuss how the congruent number problem can be solved given a weak form of the Birch and Swinnerton-Dyer conjecture (Clay Mathematics Institute Prize problem).  Most of this talk will be widely accessible, with the exception of a few technical details.