Finiteness of Mordell-Weil

Bryden Cais and Kris Klosin

These are notes for a talk given in the 2003-2004 VIGRE Number Theory Working Group. The goal of that seminar was to understand Mazur's fundamental paper Modular Curves and the Eisenstein Ideal.

The goal of these notes is to show that the modular jacobian J_0(N)/Q has a nontrivial quotient of rank 0 for any prime N such that the genus of X_0(N) is positive.