CoCalc Shared Fileswww / papers / motive_visibility / email-05-27-02.txt
Author: William A. Stein
1Computing ord_q(c_7(2)) on page 11 for the form 567k4L:
2
3Here q has residue characteristic 13.  On page 4 we define
4ord_q(c_7(2)) to be
5
6   length H_f^1(Qp,T_lambda(2))_tors - ord_q((1 - (a_7)/49 + 1/7).
7
8We have
9
10   49*(1 - (a_7)/49 + 1/7) = 49 - a_7 + 7.
11
12Also, by looking at the characteristic polynomial of a_7, we see that
13a_7 = 7 (mod 13).  Thus
14
15        49 - a_7 + 7 = 49 - 7 + 7 = 49 =/= 0 (mod 13),
16
17so ord_q((1 - (a_7)/49 + 1/7) = 0, and
18
19   ord_q(c_7(2)) = length H_f^1(Qp,T_lambda(2))_tors.
20
21I don't know what that length is, but I bet you can figure it out...
22
23----------------------------------------------------
24
25Here are some comments about small things that we might change in the
26paper (I can edit my copy and send it to you, or you can edit yours,
27whichever you prefer).  Let me know what you think.
28
29 * Page 1, Paragraph 1: add "of $E$." at the end of the paragraph.
30
31 * Page 1, Paragraph 2: replace "elements of order $m$." by
32   "elements of prime order $m$", because Cremona and Mazur only
33   do what they do for $m$ prime.  For example, if $m=15$, they
34   would treat $3$ and $5$ separately using different elliptic
35   curves.
36
37 * Page 2, Paragraph 2 (first new paragraph): replace "we are unable
38   to predict the exact order of Sha" with "we are unable to compute
39   the exact order of Sha predicted by the Bloch-Kato conjecture."
40
41 * Page 3, second and third paragraph:  "The length of its
42   lambda-component ... which we call #Sha(j)."  Do we mean
43   #Sha(j)[\lambda^{\infty}]?  Doesn't the "#Sha(j)" that we
44   just defined depend on lambda?  Also, I'm not sure I like
45   #'s for exponents, since I always use # for cardinality.
46
47 * Page 3, change "do this in any way such that" to
48   "do this in a way such that", since we don't do it in every
49   possible way such that...
50
51 * Page 4, In the statement of Bloch-Kato.  I'd like to move the
52   text "The above formula is to be interpreted as an equality of
53   fractional ideals of E.  (Strictly speaking ... E=Q.)" closer
54   to the formula, if possible.
55
56 * Page 4, near bottom: "As on p. 30 of ..." should be
57   "As on p.~30 of ..." (as it is, LaTeX think "p." is the end
58   of one sentence and "30 of" the beginning of another, so it
59   includes excessive space.
60
61 * Page 6, third line: For clarity, put parenthesis around the
62   denominator (2*pi*i)^(k/2)*Omega.  Otherwise the expression
63   would mean Omega*L(f,k/2)/(2*pi*i)^(k/2) to any calculator.
64   I.e., we need paranthesis because multiplication and division
65   by convention have the same precedence.
66
67 * Page 7, Beginning of Section 6.  We should say what f and g
68   are, i.e., that they are exactly as at the beginning of section 5.
69
70 * Page 7, statement of Theorem.  We never say what $w_p$ is.
71   We should say, write before the statement of the theorem, that
72   $w_p$ is the common eigenvalue of the Atkin-Lehner involution
73   $W_p$ on $f$ and $g$.  Also, the sentence that contains $w_p$
74   in the statement of the theorem is complicated and hard for
75   me to understand.
76
77 * Page 8, First paragraph of proof.  I don't understand this at all.
78   What short exact sequence are we using?  We should say.  I guess
79   it is (a twist of)
80
81                0 --> A[q] ---> A --> A --> 0,
82
83   where it's not really clear to me what the map A --> A is.  In any
84   case, in order for H^1(Q,A[q]) to inject into H^1(Q,A), don't we
85   need H^0(Q,A)=0, not H^0(Q,A[q])=0 as our paper currently asserts?
86   (I'm ignoring twists for the moment.)  It seems to me that this is
87   where we'll use our hypothesis that L(f,k/2)=/=0.
88
89 * Page 8, (1) of proof:  "It follows from d in H^1_f..." should be
90   replaced by "It follows from our assumption that d in H^1_f..."
91                                ^^^^^^^^^^^^^^^^^^^
92
93 * Page 8, (2) of proof:  "It suffices to show that dim H^0 ..."
94   Why?  Again, what diagram is being chased?   It's somehow not
95   immediately clear to me.  If I have some idea, or if you draw
96   me a diagram with ASCII characters, I'd be glad to typeset
97   it using the xypic package.
98
99 * Page 9, "theirs forbidding $q$ from" should be replaced by
100   "theirs forbidding~$q$ from".
101
102 * Page 9, "i.e. if " should be replaced by "i.e., if ".
103
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