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