Winding Numbers
Recall the function lambda which takes a rational number, then sends it to the sum of a_n/n times e^(2pi i n r)
Now define the modular symbol [r]+ := Re(lambda(r))/Omega+, which is rational.
Theorem: lambda(0) = L(E/Q,1)
proven by Mazur-Tate-Teitelbaum in their work on the p-adic BSD conjecture.
Corollary: [0] = L(E/Q,1)/Omega+ is rational
Also we can phrase BSD over Q in terms of modular symbols.
Theorem: (Kato, Urban-Skinner, et. al.)
L(E/Q,1) nonzero implies E(Q) and Sha finite; L(E/Q,1)=0 implies one is infinite.
In fact the rational point 0 viewed as a cusp is in X0(N)(Q), so phi(0) is a torsion point of E, so the denominator of [0] is a divisor of c*#E(Q) (i.e. bounded, and usually quite small)
Using the modular symbols one can find the size of Sha when finite.
Abelian Fields
let K be an abelian extension of Q; call m, the minimal integer such that Q(zeta_m) contains K, the conductor of K. Then there is a surjection from Z/mZ* onto the galois group G of K/Q.
Now we can view any character from G to C* as a Dirichlet character from Z to C by associating elements of G with their preimages in the above map which are clearly indexed by integers mod m*.
We now introduce three hypotheses.
I: no additive place ramifies in K/Q. This is quite necessary, or the results become false.
II: K is totally real. This is to avoid needing to introduce and use complex modular symbols.
III: d=[K:Q] is coprime to m. This is rarely used and simply excludes Iwasawa theory.
Now we will define and discuss Stickelberger elements of formal groups over all abelian extensions of Q, which nearly form an Euler system.
Define theta_E/K as the sum over a mod m* of [a/m]sigma_a in Q[G] where G = Gal(K/Q).
Lemma: let l be a prime not dividing m (here we use III)
define the Norm map N_L/K: Q[Gal(L/Q)] -> Q[Gal(K/Q)] by reducing the actions of Z/lmZ* to an action of Z/mZ*.
N_L/K(theta_E/L) is sigma l inverse times the sum (sigma l minus a_l plus delta_N(l) times sigma l inverse) times theta_E/K; where delta is 1 if l does not divide N and 0 if it does (N here is the conductor of E)
This essentially defines an Euler system, except from the factored sigma l inverse. So when that map is trivial, we do get an Euler system.
The proof of this lemma essentially involves pushing around the formulas for modular symbols and the function lambda. This is particularly exciting because Euler systems have deep structure that allows one to prove many things, but also because there are around 4 actual existing examples of Euler systems.
