> Looking again at the description below, I think it's clear that the > algebraic parts you calculate are the norms of the ones appearing in > our paper (at least up to divisors of k!N\phi(N)). I assume this is > the method used to calculate the "L(A,2)/ \Omega_A" referred to in > Section 2.1 of something dated August 29th 2000 (William, the thing > you gave me in Nottingham, with the table of examples). I don't see whether or not this is true. The definition of Omega_f^{-} in our paper is unclear to me, in the sense that I don't see how one might actually compute it. Your Omega_f^{-} isn't well-defined (it depends on a choice of delta), so I just don't know whether what I call L(A,2)/Omega_A^{-} is the norm of what you would call L(f,2)/Omega_f^{-}. In any case, for the purposes of "Constructing Visible Sha" I don't think we need to know anything about L(A,2)/Omega_A^{-}. The only reason we want to know about L(A,2)/Omega_A^{-} is to conclude that the Bloch-Kato conjecture is making predictions that agree with what we prove. But you amazingly deduce most of this in Section 5. However... > are predicted by Bloch-Kato. Section 5 of our paper only proves that q > divides it, not q^2, so I'll put something in Section 7 about this, Ahh, except for the square bit. So, if we understood the relationship between L(A,2)/Omega_A^{-} and L(f,2)/Omega_f^{-}, then maybe we could deduce that q^2 divides L(f,k/2)/vol_oo in our examples. Again, it's not clear to me what the relationship is. > expanding on the comment at the end of the introduction. In each case, > it is really a certain degree-1 prime divisor Q of q which we are > interested in. If the other prime divisors of q have degree at least 3 > then we lose nothing by passing to the norm, and know that our > algebraic part is exactly divisible by Q^2. This is except in those > cases where q^4 divides the norm, like 581k4, where we need the other > factors to have degree 5 or more, but 3 or more would be enough to > ensure that the congruences of modular forms are mod Q^2 rather than > involving more than one prime divisor of q, or a degree 2 divisor. Do > the other divisors of q always have large degree? I'm going to officially assume that L(A,k/2)/Omega_A^{+/-} equals Norm(L(f,k/2)/Omega_f^{+/-}) for the rest of this discussion. In the first example, 127k4C, the factors of (43) in the degree-17 field have degrees 1, 3, 4, and 9. By section 5 of our paper, don't we know that q | L(A,k/2)/Omega_A^{+/-}? If q^2 || L(A,k/2)/Omega_A^{+/-}, then Q.R || L(f,k/2)/Omega_f^{+/-} where Norm(R) = q. So if Q is the only prime over (q) of degree 1, then R must equal Q. Thus I think that one needs that the other primes have degree at least 2 instead of degree at least 3. Again, I don't know whether this holds in all of our examples, but I could check it. Should I? > I think the only other thing that needs sorting out before our paper > is ready is my footnote 2 on page 11. Do you have the Fourier > coefficients for these forms so that this can be checked? I don't immediately know how to find the local L-factor at bad primes for forms of higher weight (I know the definition, but not explicitly enough to immediately compute it). Do you need anything more than the Fourier coefficient of 567k4L at p=7? It satisfies the polynomial f7 = x^12 - 84*x^11 + 3234*x^10 - 75460*x^9 + 1188495*x^8 - 13311144*x^7 + 108707676*x^6 - 652246056*x^5 + 2853576495*x^4 - 8877793540*x^3 + 18643366434*x^2 - 23727920916*x + 13841287201 = (x-7)^12 (mod 13). Thus a_7(f) = 7 (mod 13). I have an idea of what happens for weight 2 modular forms for Gamma_0(N) in the level-raising context of Section 7.4. I think that the order of the *geometric* component group at p (=7 in this example) is always divisible by Q (|13 in this example). However, the Tamagawa number at p is either a power of 2 or definitely divisible by Q, depending on the sign of the Atkin-Lehner involution W_7 (since Frobenious acts on the component group through -W_7). If g has sign +1 (like in our example) and the sign of W_7 is +1, then Q doesn't divide the Tamagawa number at p. If W_7 = -1, then I think it definitely does. We have W_7 = -1 for 567k4L, so I'd *guess* by analogy with the weight 2 situation that ord_q(c_7(2)) > 0. Are these Bloch-Kato Tamagawa numbers orders of finite groups fixed under the action of -AtkinLehner? (Maybe the groups we take the length of on page 4?) The wording of the first sentence of the second paragraph of Section 7.4 is confusing because it suggests that the relative levels of f and g has something to do with the signs in the functional equation. What do you think? William