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Author: William A. Stein
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Computing ord_q(c_7(2)) on page 11 for the form 567k4L:
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Here q has residue characteristic 13. On page 4 we define
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ord_q(c_7(2)) to be
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length H_f^1(Qp,T_lambda(2))_tors - ord_q((1 - (a_7)/49 + 1/7).
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We have
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49*(1 - (a_7)/49 + 1/7) = 49 - a_7 + 7.
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Also, by looking at the characteristic polynomial of a_7, we see that
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a_7 = 7 (mod 13). Thus
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49 - a_7 + 7 = 49 - 7 + 7 = 49 =/= 0 (mod 13),
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so ord_q((1 - (a_7)/49 + 1/7) = 0, and
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ord_q(c_7(2)) = length H_f^1(Qp,T_lambda(2))_tors.
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I don't know what that length is, but I bet you can figure it out...
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----------------------------------------------------
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Here are some comments about small things that we might change in the
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paper (I can edit my copy and send it to you, or you can edit yours,
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whichever you prefer). Let me know what you think.
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* Page 1, Paragraph 1: add "of $E$." at the end of the paragraph.
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* Page 1, Paragraph 2: replace "elements of order $m$." by
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"elements of prime order $m$", because Cremona and Mazur only
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do what they do for $m$ prime. For example, if $m=15$, they
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would treat $3$ and $5$ separately using different elliptic
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curves.
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* Page 2, Paragraph 2 (first new paragraph): replace "we are unable
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to predict the exact order of Sha" with "we are unable to compute
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the exact order of Sha predicted by the Bloch-Kato conjecture."
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* Page 3, second and third paragraph: "The length of its
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lambda-component ... which we call #Sha(j)." Do we mean
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#Sha(j)[\lambda^{\infty}]? Doesn't the "#Sha(j)" that we
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just defined depend on lambda? Also, I'm not sure I like
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#'s for exponents, since I always use # for cardinality.
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* Page 3, change "do this in any way such that" to
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"do this in a way such that", since we don't do it in every
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possible way such that...
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* Page 4, In the statement of Bloch-Kato. I'd like to move the
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text "The above formula is to be interpreted as an equality of
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fractional ideals of E. (Strictly speaking ... E=Q.)" closer
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to the formula, if possible.
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* Page 4, near bottom: "As on p. 30 of ..." should be
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"As on p.~30 of ..." (as it is, LaTeX think "p." is the end
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of one sentence and "30 of" the beginning of another, so it
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includes excessive space.
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* Page 6, third line: For clarity, put parenthesis around the
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denominator (2*pi*i)^(k/2)*Omega. Otherwise the expression
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would mean Omega*L(f,k/2)/(2*pi*i)^(k/2) to any calculator.
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I.e., we need paranthesis because multiplication and division
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by convention have the same precedence.
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* Page 7, Beginning of Section 6. We should say what f and g
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are, i.e., that they are exactly as at the beginning of section 5.
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* Page 7, statement of Theorem. We never say what $w_p$ is.
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We should say, write before the statement of the theorem, that
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$w_p$ is the common eigenvalue of the Atkin-Lehner involution
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$W_p$ on $f$ and $g$. Also, the sentence that contains $w_p$
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in the statement of the theorem is complicated and hard for
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me to understand.
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* Page 8, First paragraph of proof. I don't understand this at all.
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What short exact sequence are we using? We should say. I guess
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it is (a twist of)
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0 --> A[q] ---> A --> A --> 0,
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where it's not really clear to me what the map A --> A is. In any
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case, in order for H^1(Q,A[q]) to inject into H^1(Q,A), don't we
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need H^0(Q,A)=0, not H^0(Q,A[q])=0 as our paper currently asserts?
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(I'm ignoring twists for the moment.) It seems to me that this is
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where we'll use our hypothesis that L(f,k/2)=/=0.
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* Page 8, (1) of proof: "It follows from d in H^1_f..." should be
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replaced by "It follows from our assumption that d in H^1_f..."
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^^^^^^^^^^^^^^^^^^^
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* Page 8, (2) of proof: "It suffices to show that dim H^0 ..."
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Why? Again, what diagram is being chased? It's somehow not
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immediately clear to me. If I have some idea, or if you draw
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me a diagram with ASCII characters, I'd be glad to typeset
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it using the xypic package.
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* Page 9, "theirs forbidding $q$ from" should be replaced by
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"theirs forbidding~$q$ from".
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* Page 9, "i.e. if " should be replaced by "i.e., if ".
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