Galois Representations and the Modularity Theorem
I will continue Katy's talk from last week, and talk about chapter 9 of Diamond and Shurman's "A First Course in Modular Forms." We'll define Galois representations corresponding to both elliptic curves and modular forms. Then we'll state the modularity theorem, which asserts that every elliptic curve corresponds to a modular form, and their Galois representations are equal. If we have time, we'll relate this back to the version of the theorem from last week, which equated points on an elliptic curve to Fourier coefficients of a modular form. The proof of the theorem, however, will be left as an exercise to the listener.