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Computing ord_q(c_7(2)) on page 11 for the form 567k4L:12Here q has residue characteristic 13. On page 4 we define3ord_q(c_7(2)) to be45length H_f^1(Qp,T_lambda(2))_tors - ord_q((1 - (a_7)/49 + 1/7).67We have8949*(1 - (a_7)/49 + 1/7) = 49 - a_7 + 7.1011Also, by looking at the characteristic polynomial of a_7, we see that12a_7 = 7 (mod 13). Thus131449 - a_7 + 7 = 49 - 7 + 7 = 49 =/= 0 (mod 13),1516so ord_q((1 - (a_7)/49 + 1/7) = 0, and1718ord_q(c_7(2)) = length H_f^1(Qp,T_lambda(2))_tors.1920I don't know what that length is, but I bet you can figure it out...2122----------------------------------------------------2324Here are some comments about small things that we might change in the25paper (I can edit my copy and send it to you, or you can edit yours,26whichever you prefer). Let me know what you think.2728* Page 1, Paragraph 1: add "of $E$." at the end of the paragraph.2930* Page 1, Paragraph 2: replace "elements of order $m$." by31"elements of prime order $m$", because Cremona and Mazur only32do what they do for $m$ prime. For example, if $m=15$, they33would treat $3$ and $5$ separately using different elliptic34curves.3536* Page 2, Paragraph 2 (first new paragraph): replace "we are unable37to predict the exact order of Sha" with "we are unable to compute38the exact order of Sha predicted by the Bloch-Kato conjecture."3940* Page 3, second and third paragraph: "The length of its41lambda-component ... which we call #Sha(j)." Do we mean42#Sha(j)[\lambda^{\infty}]? Doesn't the "#Sha(j)" that we43just defined depend on lambda? Also, I'm not sure I like44#'s for exponents, since I always use # for cardinality.4546* Page 3, change "do this in any way such that" to47"do this in a way such that", since we don't do it in every48possible way such that...4950* Page 4, In the statement of Bloch-Kato. I'd like to move the51text "The above formula is to be interpreted as an equality of52fractional ideals of E. (Strictly speaking ... E=Q.)" closer53to the formula, if possible.5455* Page 4, near bottom: "As on p. 30 of ..." should be56"As on p.~30 of ..." (as it is, LaTeX think "p." is the end57of one sentence and "30 of" the beginning of another, so it58includes excessive space.5960* Page 6, third line: For clarity, put parenthesis around the61denominator (2*pi*i)^(k/2)*Omega. Otherwise the expression62would mean Omega*L(f,k/2)/(2*pi*i)^(k/2) to any calculator.63I.e., we need paranthesis because multiplication and division64by convention have the same precedence.6566* Page 7, Beginning of Section 6. We should say what f and g67are, i.e., that they are exactly as at the beginning of section 5.6869* Page 7, statement of Theorem. We never say what $w_p$ is.70We should say, write before the statement of the theorem, that71$w_p$ is the common eigenvalue of the Atkin-Lehner involution72$W_p$ on $f$ and $g$. Also, the sentence that contains $w_p$73in the statement of the theorem is complicated and hard for74me to understand.7576* Page 8, First paragraph of proof. I don't understand this at all.77What short exact sequence are we using? We should say. I guess78it is (a twist of)79800 --> A[q] ---> A --> A --> 0,8182where it's not really clear to me what the map A --> A is. In any83case, in order for H^1(Q,A[q]) to inject into H^1(Q,A), don't we84need H^0(Q,A)=0, not H^0(Q,A[q])=0 as our paper currently asserts?85(I'm ignoring twists for the moment.) It seems to me that this is86where we'll use our hypothesis that L(f,k/2)=/=0.8788* Page 8, (1) of proof: "It follows from d in H^1_f..." should be89replaced by "It follows from our assumption that d in H^1_f..."90^^^^^^^^^^^^^^^^^^^9192* Page 8, (2) of proof: "It suffices to show that dim H^0 ..."93Why? Again, what diagram is being chased? It's somehow not94immediately clear to me. If I have some idea, or if you draw95me a diagram with ASCII characters, I'd be glad to typeset96it using the xypic package.9798* Page 9, "theirs forbidding $q$ from" should be replaced by99"theirs forbidding~$q$ from".100101* Page 9, "i.e. if " should be replaced by "i.e., if ".102103104105106107108109