CoCalc Public Fileswww / papers / thesis / mazur-comments.txt
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1Dear William,
2
3   I have begun to read your thesis, and-- as you can expect-- I am very
4impressed with it!    Probably, for table 1.1 instead of the cryptic
5parenthesis "(ignoring odd parts)" it  might be good to expand the
6parenthesis a bit by saying something like
7
8(the entries in the columns "order of Sha"  and "modular degree"  are only
9the odd parts of the actual "order of Sha"  and "modular degree").
10
11In table 1.1 do you know the Mordell-Weil ranks of the abelian varieties in
12the B column?  I would imagine that the Mordell-Weil groups of these B's,
13say when tensored with the field of fractions F of the corresponding ring
14of Fourier coefficients, is always of dimension 2 over F.  If not, e.g., if
15some of them are of higher dimension, it would be worth remarking.
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17 Also, since you talk about reversing "visiblity", i.e.,since you sometimes
19table, i.e.) an abelian variety  B with positive Mordell-Weil rank to try
20to guarantee some nontrivial Sha in an abelian variety A which is congruent
21to B, one might "more systematically" try to follow through on this: Do you
22have data on how often it is the case, for example, that when you have an
23optimal abelian variety B with MW-rank 2 (over F, of course), you also get
24an A congruent to B, with some Sha in A made visible in B?  Must be very
25very often, I imagine  (especially when the level goes up).  I haven't
26found a place where you explicitly list this information.
27
28 The example of level 2333  is amazing!    Clearly this example is sitting
29around waiting for someone to "do something" with it... but what?
30
31  The cases 1483, 1567, 2029, and 2593?   are also interesting because, as
32I gather from the table (perhaps you should say explicitly whether this is
33what happens...) ALL of Sha (even though it has two nontrivial p-primary
34components in each of these cases) is made visible in a single B.  I assume
35that this is true?  In contrast in the case 1913E I guess it can only be
36the 5-primary component of Sha that is visible in 1913A (?)   In any event,
37all this should be discussed more explicitly (unless yuou do this somewhere
38and I ahve missed it).
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41    More later,
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43       Barry
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