Hi Barry:

I had two questions:

(1) This question is regarding the proof of visibility in either of the two examples of 571A1 or 681B1 on p. 22. Let K = Q, E = 571A1 (or 681B1), A = F = 571B1 (or 681C1 respectively), p = 2 (or 3 respectively), Φ_{E} = E[p] and Φ_{A} = A[p] (I am using notation similar to that on p. 19). From the discussion on p. 22, we have an isomorphism i : E[p] ≅ A[p] that is Galois-equivariant, and assuming BSD, we know � (E/Q) is isomorphic to Z/pZ × Z/pZ. Let σ be an element of � (E/Q) ⊆ H^{1}(G_{Q}, E); we want to show that it is visible in J = (A ⊕ E)/{A[p] = E[p]}.

For this we will use the argument on p. 19 (starting with the line “There is a converse...” and up to the paragraph beginning “Referring to our list...”). The conditions (i), (ii), and (iii) on p. 19 are already satisfied. Let h be an element of H^{1}(G_{Q}, E[p]) mapping to σ (such an h exists by the long exact sequence of cohomology; this verifies hypothesis (a) on p. 19). Next we need to check hypothesis (b) on p. 19. From p. 22, assuming BSD, we also know that � (A/Q) is trivial. Thus if we knew that i(h) ∈ H^{1}(G_{Q}, A) actually lies in � (A/Q) ^{1} (which is contained in H^{1}(G_{Q}, A)), then hypoth- 1 esis (b) on p. 19 would be verified, and by the argument on p. 19, σ would be visible in J.

But we do not know offhand that i(h) lies in � (A/Q) – for that we have to check local triviality of the torsor i(h). I think that under some additional hypotheses (which can be verified), this would follow from the argument involving flat cohomology that you explained to William and myself; but that is not mentioned in your paper, so I suspect more needs to be said to prove visibility in your paper.

(2) In the last paragraph of p. 22, you say that the two generators of the Mordell-Weil group of A = 571B1 (or 681C1 respectively) explain the two generators of � (E/Q). But by definition, as on p. 16, the elements of � (E/Q) are explained by elements of B(Q), where B = J/E. Now A is not quite isomorphic to B, although it is isogenous to it. In any case, if P is a generator of A(Q), then the image of (P, 0) in J(\overline{Q}) actually lies in J(Q), and hence maps to 0 in H^{1}(G_{Q}, E) (by the part J(Q) → B(Q) → H^{1}(G_{Q}, E) of the long exact sequence corresponding to the short exact sequence 0 → E → J → B → 0, as on p. 16).

What one has to do instead is to find a point Q in J(\overline{Q}) such that pQ = P (such a Q exists, of course) and then the image of (Q, 0) in B(\overline{Q}) actually lies in B(Q) (since ∀σ ∈ G_{Q}, Q^{σ} − Q ∈ E[p], and hence the image of Q^{σ} − Q in B = J/E is 0), and this image of (Q, 0) in B(Q) explains an element of H^{1}(G_{Q}, E)[p]. This is without assuming BSD, but of course we still have to prove that we got a non-trivial ^{2} element of H^{1}(G_{Q}, E)[p], and 2 further that it lies in � (E/Q) (which may again require some hypotheses as in question (1) above).

1Will, I think this is where you had to use the extra hypothesis in your theorem. Is there a problem for p = 2 (the level 571 case)?

In any case, I think the statement on p. 22 that the two generators of the Mordell-Weil group of 571B1 explain the two generators of � (571A1/Q) needs some clarification.

Please do let me know if I made a mistake above!

Sincerely,

Amod

2Will, this corresponds to the injectivity in what you prove. Is there a problem at p = 2 (for the 571 case)?