Chris Second Talk

Chris' second talk, from Tuesday morning (it is now Thursday afternoon?! I guess I'll be writing these up during my vacation rather than keeping up with them daily)

The talk started by defining a function lambda from rational numbers to C, but taking the sum of a_n/n times e^(2pi i n r) where a_n are the coefficients from the Fourier expansion of the L-function of an elliptic curve over Q.

Then it turns out (Due to Manin and Drinfield) that the image of lambda is in 1/t times a lattice for some t, where that lattice is the image of H1(E(C),Z) (the free groups of paths on points of E(C)) under the map of integration of the invariant differential along those paths, extended linearly.

A bunch of other stuff happened (mostly formulas which are used in the proofs of the following) but I'm not going to copy it down and I'm skipping a bit, to a point where we're considering a map phi from the modular curve X0(N) to our elliptic curve E, and the differential w_x which is the pullback of the invariant differential on E under phi.

then the integral of w_x along a path from a rational point r to p*r in X0(N) is 1/c times the integral of w along the image of that path in E, which will be a loop, specifically it is in 1/ct Lambda, the where Lambda with a capital L is the lattice from earlier.

A corollary of this is that the images of the rational cusps of X0(N) in E are torsion points, since the integral of ct[phi(r)] is in Lambda.

Chris followed with an unpublished proposition (!) which was that for any elliptic curve one can find an isogenous curve such that t is coprime to all odd primes of semistable reduction.

Now we define modular symbols, which are functions from Q to Q coming from an elliptic curve.

[r] := Re(lambda(r))/Omega+

recall (though I might not have typed it up) that Omega+ is the positive integral of w_x along the curve from i*infinity to a generator (the one that makes it positive) of the +1 eigenspace of the action of the Galois group on the lattice Lambda. Technically we should define this as [r]+ and do something similar for [r]- for when we're in an appropriate imaginary field but we will be working in totally real fields so we won't worry about that.

Now notice that [-r]=[r] and [r+1]=[r]. In the future we generally deal with modular symbols in the context of sums of [a/m] where a ranges through the group of units modulo m.

In the next talks things get more interesting; there's a lot of definitions here and a lot are somewhat complicated.