Dear William,
I have begun to read your thesis, and-- as you can expect-- I am very
impressed with it! Probably, for table 1.1 instead of the cryptic
parenthesis "(ignoring odd parts)" it might be good to expand the
parenthesis a bit by saying something like
(the entries in the columns "order of Sha" and "modular degree" are only
the odd parts of the actual "order of Sha" and "modular degree").
In table 1.1 do you know the Mordell-Weil ranks of the abelian varieties in
the B column? I would imagine that the Mordell-Weil groups of these B's,
say when tensored with the field of fractions F of the corresponding ring
of Fourier coefficients, is always of dimension 2 over F. If not, e.g., if
some of them are of higher dimension, it would be worth remarking.
Also, since you talk about reversing "visiblity", i.e.,since you sometimes
start with an abelian variety (of "type B", to follow the notation of your
table, i.e.) an abelian variety B with positive Mordell-Weil rank to try
to guarantee some nontrivial Sha in an abelian variety A which is congruent
to B, one might "more systematically" try to follow through on this: Do you
have data on how often it is the case, for example, that when you have an
optimal abelian variety B with MW-rank 2 (over F, of course), you also get
an A congruent to B, with some Sha in A made visible in B? Must be very
very often, I imagine (especially when the level goes up). I haven't
found a place where you explicitly list this information.
The example of level 2333 is amazing! Clearly this example is sitting
around waiting for someone to "do something" with it... but what?
The cases 1483, 1567, 2029, and 2593? are also interesting because, as
I gather from the table (perhaps you should say explicitly whether this is
what happens...) ALL of Sha (even though it has two nontrivial p-primary
components in each of these cases) is made visible in a single B. I assume
that this is true? In contrast in the case 1913E I guess it can only be
the 5-primary component of Sha that is visible in 1913A (?) In any event,
all this should be discussed more explicitly (unless yuou do this somewhere
and I ahve missed it).
More later,
Barry