Course 18.784: Seminar in Number Theory


This will be an introduction to the basics of modular forms, with talks given by students. I plan to start with the hyperbolic and complex geometry of the upper half plane, and move on to some examples (theta functions, Eisenstein series) and fundamental theorems. Depending on pace, we may consider some more advanced topics, such as positive characteristic questions, modular jacobians and representations of SL2(R).


(4/14) The schedule of final talks is up. I will put up some notes on what I expect from a final paper.

(3/2) The definition of the lattice IIp,q given in class was incorrect (and it was entirely my fault), so here is the correct definition. The lattice vectors have the form x1,...,xp,y1,...,yq, where the sum of coordinates is even, and either all xi and yj are integers or all are integers plus 1/2. Norms are given by x12+...+xp2-y12-...-yq2. When p is congruent to q mod 8, the resulting lattice is even and unimodular.