William Stein, U.C. Berkeley

``The Modular Degree, Component Groups, and the Denominator of L(Af,1)/Omega''

March 10, 1999

Let f be a weight 2 newform and A_f the corresponding optimal quotient of J_0(N). This talk will consist of:

I. DENOMINATORS: I will give a short proof that, aside from the Manin constant, the odd part of the denominator of L(A_f,1)/Omega divides
gcd { #A_f(F_p) : p does not divide 2N }.

This can be considered as evidence for the Birch and Swinnerton-Dyer conjecture.

II. MODULAR DEGREE: Next I will talk about the "generalized modular degree" of A_f. Autoduality of J_0(N) gives rise to a natural isogeny delta: dual(A_f) -----> A_f By working over the complex numbers and using the period map it is possible to find a simple explicit formula for Ker(delta), reminiscent of the formula we have for L(A_f,1)/Omega. Namely, Ker(Delta) = [ Phi(H_1(X_0(N),Z)) : Phi(H_1(X_0(N),Z)[p_f]) ] where p_f is the annihilator of f in the Hecke algebra and Phi is the period map associated to A_f. I will try to prove that this formula is correct, and give some interesting numerical examples in which Ker(Delta) is divisible by surprisingly large primes.

III. COMPONENT GROUPS: Assume p exactly divides N. Following Ribet's letter to Mestre I will derive a formula for the number of connected components c_p of A_f in terms of character groups (Ribet did this when dim A_f = 1). I know how to compute explicitly all but one term in the formula, namely the order of the cokernel of the map from the character group of the torus of A_f to that of its dual. Hopefully my inability to compute this quantity is simply a reflection of my ignorance about the torsion on the Neron model mod p, so if you're swift with group schemes perhaps you can understand this term, and obtain an algorithm for computing c_p(A_f)!