BSD Summer School Talk 2

I've split things up by talk because I want the posts to each be a manageable size and not to have to write too much at once.

Chris' first talk: Modular Symbols

The aim here is to find out about BSD over number fields and their relationship to BSD over Q.

In general we will fix E/Q with a global minimal model and the Neron differential (which is the standard invariant differential).

We will consider H1(E(C),Z) the group of loops in E(C) based at 0 modulo homotopy, which since E(C) is a torus is isomorphic to Z^2. There is a map called the period map from here to C, which sends a loop to the integral of the invariant differential modulo about that loop. Its image is a lattice in C called the Neron lattice, and E(C) is isomorphic to C mod that lattice.

X0(N) is the modular curve of level N, which over C is the upper half plane adjoin the point at infinite and Q, modulo \lambda_0(N), the group of matrices in SL2(Z) which are upper triangular mod N.

Theorem due to Wiles, Taylor, et. al.:

There is a non-constant morphism from the modular curve of level N to an elliptic curve of conductor N sending the point at infinity to the zero of E

There exists an integer, the Manin constant, such that the pullback of the invariant differential of E is a scalar multiple of gamma_x, given by an explicit formula with coefficients closely related to the Taylor expansion of the L function of E/Q.

The scalar multiple above can be made to be 1 by moving to an isogenous curve.

Our next purpose is to show that the special values of L-functions are rational multiples of the positive generator of the subgroup of eigenvectors of the period map with eigenvalue 1.

Calling the images of Q and infinity in X0(N)(C) cusps, we take the free group on paths between cusps modulo homotopy (when tensored with Qp this gives us the dual of etale cohomology of the modular curve over Qp)

We now take the map lambda to be the integral over these paths on X0(N)(C) of the pullback of the invariant differential. Because of our explicit formula for that differential, we can integrate; though this gives us a series that converges very slowly and not obviously.

Hm. Rereading this, I think I'm understanding more of what is going on, but it doesn't feel very well structured; goals were outlined but not really achieved.