BSD Plan for 2007-04-09

1. Define L_p(E,T) for p odd good ordinary prime.
   Let alpha be the root of x^2-a_px+p with |alpha| = 1.
    (0) Recall definition:
          [r] = 2*pi*i/Omega_E * (...)

    (a) define a measure on Z_p^*:
        mu_E(a+p^n*Z_p) = 1/alpha^n*( [a/p^n] - [p*a/p^n]/alpha )

    (b) p-adic L-series is a function on characters
                   Hom(Z_p^*, C_p^*)
        given by sending chi to
             L_p(E,chi) = integral chi * dmu,
        whatever that means (it's defined via Riemann sums).

    (c) Fix the topological generator gamma=1+p of 1+pZ_p, so a 
        character in Hom(1 + p*Z_p, C_p^*) is given by choosing
        any u in
              D = { u in C_p : |u-1|_p < 1 } 
        and sending gamma=1+p to it.    Let psi_u be that character.
        For x in Z_p^*, let <x> = x / teich(x mod p). 
        For any u in D, define character chi_u on Z_p^* by 
               chi_u(x) = psi_u(<x>)
        Define a map f: D --> C_p by
             u |---> L_p(E,chi_u).
        FACT: f is a p-adic analytic function.
        Let L_p(E, T) in Q_p(alpha)[[T]] be the Taylor expansion 
        about u=1; this converges { z in C_p : |z|_p < 1 }.
        We have for u in D that
          L_p(E, u-1) = L_p(E, chi_u).

    (d) Riemann sums approximation:
        The n-th approximation to L_p is
           P_n(T) = sum(j=0,p^(n-1)-1,
                       sum(a=1,p-1,
                            mu(teich(a)*gamma^j + p^n*Z_p))*(1+T)^j)
           in Q_p(alpha)[T]
        and 
           lim_{n-->oo} P_n(T) = L_p(E,T).

    (e) Interpolation property: 

    (f) Suppose zeta_j is a primitive p^j-th root of unity.  Then
             P_n(zeta_j - 1) is the same for all n > j.
        Thus P_{n+1} - P_n vanishes at zeta_j - 1 for j < n, so
        P_{n+1} - P_n is divisible by 
                     omega_n(T) = (1+T)^{p^n} - 1.

        
2. Student feedback: 
    1. What was clear?
    2. What was murky?
    3. What should I change to get a perfect evaluation from you?