CoCalc Shared Fileswww / papers / motive_visibility / email-05-24-02.txt
Author: William A. Stein
1> Looking again at the description below, I think it's clear that the
2> algebraic parts you calculate are the norms of the ones appearing in
3> our paper (at least up to divisors of k!N\phi(N)).  I assume this is
4> the method used to calculate the "L(A,2)/ \Omega_A" referred to in
5> Section 2.1 of something dated August 29th 2000 (William, the thing
6> you gave me in Nottingham, with the table of examples).
7
8I don't see whether or not this is true.  The definition of
9Omega_f^{-} in our paper is unclear to me, in the sense that I don't
10see how one might actually compute it.  Your Omega_f^{-} isn't
11well-defined (it depends on a choice of delta), so I just don't know
12whether what I call L(A,2)/Omega_A^{-} is the norm of what you would
13call L(f,2)/Omega_f^{-}.
14
15In any case, for the purposes of "Constructing Visible Sha" I don't
16think we need to know anything about L(A,2)/Omega_A^{-}.  The only
17reason we want to know about L(A,2)/Omega_A^{-} is to conclude that
18the Bloch-Kato conjecture is making predictions that agree with what
19we prove.  But you amazingly deduce most of this in Section 5.
20However...
21
22> are predicted by Bloch-Kato. Section 5 of our paper only proves that q
23> divides it, not q^2, so I'll put something in Section 7 about this,
24
25Ahh, except for the square bit.  So, if we understood the relationship
26between L(A,2)/Omega_A^{-} and L(f,2)/Omega_f^{-}, then maybe we could
27deduce that q^2 divides L(f,k/2)/vol_oo in our examples.   Again, it's
28not clear to me what the relationship is.
29
30> expanding on the comment at the end of the introduction. In each case,
31> it is really a certain degree-1 prime divisor Q of q which we are
32> interested in. If the other prime divisors of q have degree at least 3
33> then we lose nothing by passing to the norm, and know that our
34> algebraic part is exactly divisible by Q^2. This is except in those
35> cases where q^4 divides the norm, like 581k4, where we need the other
36> factors to have degree 5 or more, but 3 or more would be enough to
37> ensure that the congruences of modular forms are mod Q^2 rather than
38> involving more than one prime divisor of q, or a degree 2 divisor.  Do
39> the other divisors of q always have large degree?
40
41I'm going to officially assume that L(A,k/2)/Omega_A^{+/-} equals
42Norm(L(f,k/2)/Omega_f^{+/-}) for the rest of this discussion.
43
44In the first example, 127k4C, the factors of (43) in the degree-17
45field have degrees 1, 3, 4, and 9.
46
47By section 5 of our paper, don't we know that q | L(A,k/2)/Omega_A^{+/-}?
48If q^2 || L(A,k/2)/Omega_A^{+/-}, then Q.R || L(f,k/2)/Omega_f^{+/-}
49where Norm(R) = q.  So if Q is the only prime over (q) of degree 1,
50then R must equal Q.  Thus I think that one needs that the other primes
51have degree at least 2 instead of degree at least 3.   Again, I don't
52know whether this holds in all of our examples, but I could check it.
53Should I?
54
55> I think the only other thing that needs sorting out before our paper
56> is ready is my footnote 2 on page 11. Do you have the Fourier
57> coefficients for these forms so that this can be checked?
58
59I don't immediately know how to find the local L-factor at bad primes
60for forms of higher weight (I know the definition, but not explicitly
61enough to immediately compute it).  Do you need anything more than the
62Fourier coefficient of 567k4L at p=7?  It satisfies the polynomial
63
64f7 = x^12 - 84*x^11 + 3234*x^10 - 75460*x^9 + 1188495*x^8 -
65     13311144*x^7 + 108707676*x^6 - 652246056*x^5 + 2853576495*x^4 -
66     8877793540*x^3 + 18643366434*x^2 - 23727920916*x + 13841287201
67    = (x-7)^12 (mod 13).
68
69Thus a_7(f) = 7 (mod 13).
70
71I have an idea of what happens for weight 2 modular forms for
72Gamma_0(N) in the level-raising context of Section 7.4.  I think that
73the order of the *geometric* component group at p (=7 in this example)
74is always divisible by Q (|13 in this example).  However, the Tamagawa
75number at p is either a power of 2 or definitely divisible by Q,
76depending on the sign of the Atkin-Lehner involution W_7 (since
77Frobenious acts on the component group through -W_7).  If g has sign
78+1 (like in our example) and the sign of W_7 is +1, then Q doesn't
79divide the Tamagawa number at p.  If W_7 = -1, then I think it
80definitely does.  We have W_7 = -1 for 567k4L, so I'd *guess* by
81analogy with the weight 2 situation that ord_q(c_7(2)) > 0.  Are these
82Bloch-Kato Tamagawa numbers orders of finite groups fixed under the
83action of -AtkinLehner?    (Maybe the groups we take the length of
84on page 4?)
85
86The wording of the first sentence of the second paragraph of Section
877.4 is confusing because it suggests that the relative levels of f and
88g has something to do with the signs in the functional equation.  What
89do you think?
90
91William
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