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