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.
    (*) Recall definition:
          [r] = 2*pi*i/Omega_E * (...)
    (*) define a measure on Z_p^*:
        mu_E(a+p^n*Z_p) = 1/alpha^n*( [a/p^n] - [p*a/p^n]/alpha )
    (*) L_p(E,T) in Q_p[[T]] is power series of function
         u |--> L_p(E,chi_u)
        about u = 1 in C_p.

2. Riemann sums
    (*) Riemann sums approximation:
        The n-th approximation to L_p is
           L_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[T]
        and 
           lim_{n-->oo} L_n(T) = L_p(E,T).

    (*) Prove that limit exists and equals p-adic L-function
        at infinitely many points. 
        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.

3. Examples.
   

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