These are Bjorn's comments on my thesis.
Each is answered with either
  "A:" or "S:"
 "A:" means the answer
 "S:" means I'm stumped at present (possibly because of lack of net access)

P: is a pointer to where I'm at.

Date: Sun, 26 Mar 2000 11:53:30 -0800 (PST)
To: was@math.berkeley.edu
Subject: chapters 1,2 of your thesis

Dear William:

So far I've read through the end of Chapter 2 in your thesis.
It's really very well written.  I must say, however, that the
technical nature of Chapter 2 made me want to skim through it
rather than read every detail; I suppose that's inevitable.

Anyway, here are the comments I have so far.  You can choose to
ignore most of them if you want; there are very few that are substantial.
I hope you won't be offended if I sometimes complain about grammar!

--Bjorn

p.2, first sentence: if I personally were asked to name the main
outstanding problem in the arithmetic of elliptic curves, I would say
it is the problem of whether there is an algorithm to compute
Mordell-Weil ranks (or equivalently, via descent, the problem of determining
whether a genus 1 curve over a global field has a rational point).
Of course this is related to BSD, and in particular is implied by
the finiteness of Sha, but to me the latter problems are secondary.

A: I changed the wording slightly, and added references to two papers
that agree with my opinion.   

p.2, Def 1.1: do you want to require that A be simple over Q? (It's 
up to you.)
A: No; I'll say "simple" when needed.

p.3, line 7: longterm should be long-term
A: OK.

p.3, Cor 1.4: you could replace "is an integer, up to a unit in"
		by $\in$
A: OK.

p.4, line -3 (i.e., 3 lines from the bottom): "1-dimensional abelian
varieties": why not call them elliptic curves
A: OK.

p.5, Thm 1.7: define $\rho_{E,p}$
A: OK.

p.5, line -3: "one expects..."  Is there some theoretical heuristic
for this?  
A: "Mazur told me so", but didn't really explain it sufficiently.
He said something about the modular degree annihilating the
"symmetric square", and the symmetric square should have nothing
a priori, to do with Sha.

   If not, it might be more accurate to write
   "numerical experiments suggest..."
   Also (to be picky), when you write "most of III(A-dual)"
  you don't really mean most of III(A-dual) for each A,
  but most as you VARY A, I am guessing.
A: Right -- thanks.  I changed it to say that
   "Numerical experiment suggests that
     as $\Adual$ varies, Sha is often not visible inside
    J_0(N).  For example ..."

p.6: "So far there is absolutely no evidence..."
I guess there is no evidence to lead one to conjecture the opposite, either.
I guess I don't understand your reasons for writing this sentence.
By writing it this way, do you mean to suggest that you are more
inclined to believe that III(A-dual) is eventually all visible
in some J_0(NM)?
A: I re-worded it to say:
 "We have been unable to compute any examples in which $\Sha(\Adual)$ is
  not visible at level~$N$, but becomes visible at some level $NM$.
  Any data along these lines would be very interesting."

p.6, next paragraph: significant difficult
A: I must have deleted this or fixed it, as I can't find it.

p.6, first sentence of 1.1.6: "...is bound to fail."
This sounds as if you've proved that it will fail.
If you haven't, maybe it would be better to say "will probably fail".
A: Thanks.

p.6, 1.1.6: give a reference for Kani's conjecture.
S: ask Frey?  ask Ernst?
--> See page 9 of Cremona-Mazur: "Kani-Shantz".

p.6, first sentence of 1.2: remove the comma
A: ok.

p.6, second sentence of 1.2: "to provably compute"
(grammatically speaking, it's incorrect to split an infinitive)
A: ok. Now the paragraph reads:
  Without relying on any unverified conjectures,
  we will use the following theorem to exhibit abelian varieties~$A$
  such that the visible part of $\Sha(A)$ is nonzero.  

p.6, Theorem 1.8: The sentence beginning "Suppose p is an odd prime..."
sounds a little funny to my ear.  I'd suggest replacing the "and" by
"or the order"
A: done.

p.6, last "sentence" of Theorem 1.8:  It's better to avoid using
a symbol as the verb of a sentence.  You could instead say
"Then there exists an injection
	B(Q)/p...."
A: good point.
  
p.7, a little over halfway down the page:
I'd suggest putting a period after "K_1=0" and then beginning a new sentence.
A: ok.

p.7, next paragraph: "the latter group contains infinitely many elements
of order p"   Maybe give a reference for this? (even though you don't
use it)
A: ok, I'll mention Shafarevich's paper.

p.7, next paragraph: archimedean
A: ok

p.7, next paragraph: define Q_v^ur.  Also, give a reference for the
generalization of Tate uniformization.
S: look at what Ken does... Faltings-Chai??

p.7, same paragraph: "it follows that there is a point Q..."
Am I missing something?
It seems to me that this doesn't work when v=p.
I'm worried...
A: It doesn't; I've weekend the statement of the theorem accordingly.
I wonder how to generalize the result to include this case...?

p.8: when you take stalks the J suddenly becomes a B! (twice)
A: woops!

p.8, middle: need a period at the end of the paragraph
A: ok

p.8, next paragaph: "The 2-primary subgroup $\Phi$ of $A \cap B$
is rational over $\Q$."
I don't see why the points in $\Phi$ have to be rational.
Oh, do you mean simply that it is rational as a subgroup?
A: I mean "rational as a subgroup", which I'll write more explicitly.

p.8, penultimate sentence of the proof: "the component group...has order
a power of 2".  In fact, it's trivial, since A-tilde has good reduction at 2.
A: Thanks for pointing this out.

p.9, line 6: quotient needs an s
A: Thanks; also, I just realized that "rank zero optimal quotients" 
should be "rank-zero optimal quotients".

p.9, two paragraphs later: "By definition, there must be other subvarieties..."
By definition of what?
A: modular degree -- I reworded this paragraph more clearly.

p.9, end of that paragraph: "can not" should be "cannot" I think.
A: I agree.

p.10, section title of 1.3.1: move "only" after "considering"
(only should be put as close as possible to the thing it is onlifying,
if you know what I mean)
A: Thanks for the tip!

p.15, section 2.1: Perhaps explain the motivation for these
definitions.  (You probably have more intuition and knowledge about
this than I do.  Is it that {a,b} was originally thought of as the
homotopy type of a path from a to b through the upper half plane (or
its projection in a modular curve).  This would explain the relations,
for instance.)
A: Yep; I'll add this.

p.16, line 2: "torsion-free quotient":  Are you claiming that this
quotient is already torsion-free, or that you are going to make it
torsion-free by dividing out its torsion subgroup if necessary?
If the latter, I think it'd be worth defining the term
"torsion-free quotient" separately.
A: I've systematically written "the largest torsion-free quotient of" in place
   of "torsion-free quotient."

p.16, line 2: You never defined Z[epsilon].
Is it the subring of the space of functions from (Z/NZ)* to C
generated by epsilon, or the subring of C generated by the values
of epsilon, or perhaps the group ring Z[G] where G is the group
generated by epsilon?  (I'm pretty sure I know the answer, but
another reader might not.)
A:  Thanks for pointing out the ambiguity.
It's the subring of C generated by the values of eps.

p.16, definition of M_k(N,epsilon;R): I think you mean
"tensor over Z[epsilon]".  "Tensor" by itself means "tensor over Z,"
which will give something very different.
A: Yep. Thanks.

p.17, first two sentences of 2.4.1: This is a little vague (and awkward).
Maybe write instead:
Let $V$ denote either a space $M_k(N,\epsilon;R)$ of modular symbols
or a space \dots of modular forms [you should clarify what sort of
spaces of forms you will consider].
The Hecke algebra $\T$ is then the subring of $\End_R(V)$
generated by the $T_n$.

Clearly T depends on the choice of N,epsilon,R.
But given this data, is it the same for modular symbols
and for modular forms?  I suppose the answer might depend on
exactly which type of modular forms you consider.

Is it obvious what the action on antiholomorphic forms is?
A: Ok.

p.17, Prop 2.7: Give a reference for this, if you're not going to prove it.
A: Ok.

p.18, line 8: it's should be its
A: Ok.

p.18, Definition 2.10: since "plus one" is acting as an adjective,
I think it'd be better to put a hyphen in the middle.
Same for minus-one.
A: Ok.

p.20, Definition 2.14: I have some questions for you: are the new and old
modular symbols disjoint?  Is their sum equal to the whole space,
or at least is their sum of finite index in the whole space?
S: I'm not sure about the sum on the Eisenstein part

p.20, Remark 2.15: "can not" should be one word I think.

p.20, Remark 2.16: Is p prime to MN?  What is F_p[epsilon]?
Is it Z[epsilon]/(p), or Z[epsilon]/(fancyp) where fancyp
is a prime of Z[epsilon] above p, or ... ?

p.20, line 4 of Remark 2.16: basis should be bases

p.20, matrix in Remark 2.16:
Are you sure you want to write it in this transposed way?
It is much more common to write linear transformations
as matrices acting on the left on column vectors.
(If you are going to keep it as is, it might help to remark
that you are doing things this way.)

p.20, bottom: It'd be better to define P^1(t) in a separate sentence.
When I first read this, I didn't realize that this was supposed to
be a definition of P^1(t) and I started looking back at earlier
pages searching for one.

p.21, top: In some sense, deterministic algorithms have a greater right
to be called algorithms than random algorithms.  Although I am sure
that from the implementation point of view it was easier to do things
the way you did them, you might at least add a comment that it is
possible to rewrite this a deterministic algorithm, say by first computing
coset representatives for Gamma(MN) in Gamma(1),...

p.21, 2.5.1: "base field"?  There has been no mention of base field
up to now, in the context of modular symbols.
Do you mean that you are now taking R to be a field?
By "degeneracy maps" do you mean alpha_t and beta_t
relativized to R?

p.22, first line of proof of Theorem 2.19: tensor over Z[epsilon] again?

p.26, middle: exists should be exist

p.27, last line of proof of Prop 2.28: "torsion free" should be torsion-free


Date: Mon, 27 Mar 2000 23:27:05 -0800 (PST)
To: was@math.berkeley.edu
Subject: more comments

Dear William:

I've finished "reading" your thesis.
Below are the rest of my comments.

--Bjorn

 p.28, diagram 2.1, etc.:  I'm really puzzled by this and your
 comment on p.44 that the degree of the composition
 	theta_f : A_f-wedge --> A_f
 need not be a square.  There's no contradiction, but there's
 a natural approach to try to prove that it IS a square,
 and I'm wondering where it goes wrong.  So here are some questions
 about the situation:
 1) For k>2 is J_k(N,epsilon) an abelian variety?
 (It was unclear to me from your remark about Shimura at the beginning
 of section 2.7 whether Shimura proved this in general or not.)

A: Shimura's construction is different; there's no reason why ours should agree.

 2) If so, is it a PPAV ? I think this is equivalent
 to the complex torus being isomorphic to its dual.

A: I think so, by properties of the Peterson inner product.

 3) If a complex torus is a quotient of an abelian variety over C,
 is it automatically an abelian variety?  (I think yes.)

A: Yes, at least in this case, because it's a quotient by an ideal of
endomorphisms.

 4) Is A_f-wedge --> J_k(N,epsilon) the map dual to J_k(N,epsilon) --> A_f ?

A: It should be, in the Shimura setting.

 5) Is theta_f always an isogeny?

A: I think so...
   Maybe your above remark can be used to show that my construction is definitely
   {\em not} Shimura's.

p.30, bottom: what does it mean to compute an O-module.
I guess what I'm really asking is, how will you present the answer?
Will you give a Z-basis?
A: I don't really care, because I'm not analyzing efficiency. 
The algorithm here takes "compute O-module" as a black box;
I think it would be a mistake for me to by precise about how this part of the
computation should go...  (I've added a paranthetical remark of this nature.)

p.31, 2nd paragraph of 3.2: In the definition of M_k(Gamma) are you
working over C?
A: fixed.
p.32, line 5: "Put R=F_p in Prop 3.6"  -- but just before Prop 3.6
you said that R was going to be a subring of C.
A: fixed.

p.32, paragraph following Lemma 3.11: a_i is an element of what?
the positive integers?
A: no, of the group it generates.

p.32, end of this paragraph: "We thus represent epsilon as a matrix"
Why call it a matrix, if it's really just a vector?
A: woops -- thanks

p.32, bottom: the ' in n'th looks a lot like an apostrophe here.
You might try $(n')^{\text{th}}$.
I personally prefer $n^{\text{th}}$ to $n$th (so much so, that I made
a macro out of it).  If \text doesn't work in your brand of tex,
try \operatorname in its place.
A: Some style manuals explicitly told me not to write n^{th}... and I agree with their argument.
I'll use paranthesis.

p.33, line before definition 3.13:
I don't understand the (2^{n-2}-1)/2.
Shouldn't it be 2^{n-3}, for n>=3 ?
A: Thanks!!!

p.33, sums in Theorem 3.14: the size and spacing of the indices of
summation looks really weird.
A: OK; I changed to summing over x in (Z/NZ)^*, which is more precise and eliminates
the need for all of the "mod"'s.

p.34: delete comma after "cumbersome"
A: I disagree.  It is an independent clause, so it is set off in commas.
I could delete "more cumbersome" and the sentence still makes sense. 
If I had said "Alternatively, a more cumbersome, way to ..." then I should have
deleted the comma.  

p.34, two lines later: "The author..."  of this thesis or of [Hij64]?
A: "of this thesis". fixed.

p.34, same sentence: "...has done this and found..."  The tenses don't
match.  How about "has done this and has found..."
A: I did it.

p.34, line -4: what is S?
A: Oops -- a space of cusp forms.

p.35, line 2 of 3.6.1: something's messed up
A: Oops -- there was an extra "\".

p.35, (3.1): this is a little weird in that M_k(N,epsilon)
is not a K-vector space
A: changed to ";K"

p.35, two lines later: how do view the elements of T
as "sitting inside M_k(N,epsilon)"?
A: I mean, as endomorphisms of ...

p.35, prop 3.15: Probably you should go back to Def 2.14
and do it over other bases, since I think here you want new
modular symbols over K, in order to get a good notion of irreducible.
A: ok.

p.35, next paragraph: "The new and old subspace of M_k(N,epsilon)^perp
are defined as in Definition 2.14."
Will the alpha_t and beta_t be replaced by beta_t^perp and alpha_t^perp,
respectively?
If so, it might be worth giving the definition in full here
rather than refer back to Def 2.14.

p.35, algorithm 3.16, lines 4-5: "Using the Hecke operators..."
Although there's nothing technically wrong with this sentence,
it tricked me into thinking it was going to be parsed differently,
if you know what I mean.  Is there some way you could rewrite it?
A: re-ordered

P: <------------------>

p.35, algorithm 3.16, 3(b): is this stated correctly?
Give a reference for the facts you are assuming, or prove them.
S: OK.  This was incorrectly and awkwardly stated. I've fixed it.
(But need to add a real reference; perhaps to Loic's paper.)

p.36, top: "repeat step 1"; do you mean just step 1, or do you
mean go back to step 1.  also, did you mean to replace p by
the next larger prime?
A: Thanks; this was ridiculous imprecise as it stood!

All over the place: Some editors consider contractions (like don't)
too informal for published math.
A: I just fixed it with grep.

p.37, alg 3.19: "Then for any randomly chosen..."
What is the mathematical meaning you have in mind here?
A: As you point out, there's no need to have the word "random".
By the way, is K infinite?
A: NOPE.


p.37, same sentence: by my convention, g(A)v is always an eigenvector;
the real question is whether it is nonzero!
A: Oops!  Your convention is my convention, too!  Thanks.  I fixed it.


p.38, alg 3.20, step 2: by Hecke operator, do you mean a T_n,
or any linear combination?  If the former, it's not obvious that
the primitive element theorem is enough.
A: Woops.  I mean the *latter*.  Not only is the former "not obvious", it isn't
true!  I found the first example in the course of my computations; it
occurs in S_2(Gamma_0(512)).  I've added a note to this effect.


p.38, step 4: "w is a freely generating Manin symbols".
Even without the "s" I'm not sure what this means.
A: a symbol of the form [P,(c,d)].

p.38, line -5: define K[f].
A: done.

p.38, bottom: Is it clear that these traces determine f uniquely?
A: No, but I'm sure it's true.
However, it is clear if we include Tr(a_n), for all n.  I've
changed to this, because it is also cleaner. 

p.39, end of 3.6: is it clear that all ties will eventually be broken?
A: I don't know.

p.41, just before def 3.25: "...we use it to computing..."
A: fixed.

p.42, line 5: "The rank of a square matrix equals the rank of its transpose..."
This holds even if the matrix is not square!
A: Ok!

p.42, first line of proof of 3.29: define O-lattice.
In particular, make clear that you insist on finite covolume.

p.42, sentence above alg 3.30: J(Q) has not been defined.
Do you mean to say that when k=2, epsilon=1,
then J can be identified with J_0(N)(C)?
A: Yep.

p.43, 3.9.1: Do you know about Glenn Steven's book,
called "Arithmetic on modular curves" or something like that?
I think maybe he works out in general over which fields
cusps on modular curves are defined.  This together with
modular symbol calculations should give a reasonable solution
to the problem.  I'm not saying that you should carry this out;
but if you feel that Steven's book is relevant, maybe you could
cite it.

A: I'm citing it.  I'll look into this in detail after I finish
my thesis.  

p.43, proof of prop 3.32:
In what space are T_p and Frob+Ver equal?
Define g.
Does all this work even in the bad reduction case?
Give reference for f(t)=x^{-g} F(x), or explain.
A: I cleaned this up a lot; strangely enough, I was giving the proof
for J_0(N) instead of A_f!  
S: I still need to give a reference for "f(t)=x^{-g} F(x)".

p.43, bottom: Are you claiming that you have a counterexample
in the form A_f ?
(By the way, you should use ; or : or . instead of ,
in the middle of this sentence.)
A: Yep!  I should state all this for general abelian varieties
$A$... and give a reference.
S: give a reference; see Cassels-Flynn...

p.44: give a reference for Prop 3.35
A: I don't know one; though, I think it is easy to prove...
   I never use this proposition, so I'll de-proposition it, and only 
   mention it in passing.

p.44, alg 3.36: define "modular kernel"
A: done.

p.46: is it known that c_A is a positive integer?
A: "YES.

p.47, second line of proof of 3.4: "of" after "smooth locus"
A: fixed, with ou.

p.47, (3.2): give a reference for the isomorphism in the middle
A: done.

p.47, middle: should Tor^1 be Tor_1 ?
A: done.

p.47: "torsion free" should be hyphenated I think (several times)
A: hmm.

p.47, line -7: Is fancyB a Neron model too?
...

p.47, same line: "In particular,..."  How does this follow from
the exact sequence from Mazur?
...

p.47, next line: why is the map on the right an isomorphism?
...

p.48, line 6: singe (I don't think that's the word you want!)
...

p.48, remark 3.44: peak (wrong word, again)

p.49, top: you define g but never use it!
A: I mean "f".

p.52, middle: "It would be interesting to know whether..."
Since you then give a counterexample, maybe it would be better
to replace "whether" by "under what circumstances".
Also, instead of saying "When k is odd this is clearly not the case"
it would be better to say "This sometimes fails for odd k"
since for some odd k and certain N it will be true (for instance
when S_k(N,epsilon) is trivial!)
A: thanks!!

p.52, line 2 of 3.13.3: "Section [AL70]"  Is this a typo?
A: oops; I used "cite" instead of "ref".

p.52, next line: missing >
A: got it.

p.53: k-2th: put k-2 in parentheses
A: Ok.

p.54, top: "to efficiently compute" split infinitive
A: *doh*

p.54, line -9: how does e_i depend on i?
A: woops.

p.55, def 3.50: "time" should be "times"
A: OK.

p.55, def 3.50: you shouldn't call it a -1 eigenspace,
since A_f(C) is not a vector space
A: Ok.

p.58, CM elliptic curves: "Let be a rational newform with complex
multiplication."  What does this mean?  Give a definition or a reference.
A: I'll give a silly, but correct, definition.

p.60, line -6: , after "purely toric" should be .
A:  thanks.

p.60, line -4: is A' the dual of A (which you later call A-wedge)?
A: yep; you guessed it, I changed my mind at some point... 

p.61, middle: on the right of the "dualize" should C be C-dual?
A: yep.

p.62, middle: the T and U are backwards in the vertical sequence.
Think of the semidirect product of G_m by G_a  (the "ax+b" group).
A: I disagree with you here.   The torus is the connected sub-thing,
not the quotient.

p.62, two lines later: remove the , after "purely toric reduction"
A: woops.

p.62, definition of X_A: should take Homs over the algebraic closure,
or else define X_A as a group scheme.
(For example, if T is a nontrivial twist of G_m, then Hom(T,G_m)=0,
which is not what you want.)
A: yep. thanks.

p.62, sequence just before 4.3: give a reference
A: ok.
S: but I should be more precise.

p.62, thm 4.2: define universal covering
A: refered to coleman's paper where everything is defined.

p.64, middle: one-motif !!!
A: got it.

p.65, example 4.7: "... is a Tate curve" over Q_p^ur.
(For a ramified extension, the answers will be different.)
A: ok.

p.65, middle: define pi_*, pi^*, theta_*, theta^*
A: ok.

p.65, bottom: prove or give a reference for the middle equality
S: I'll have to dig up a reference for this compatibility when I get back.

p.66, middle: "Suppose L is of finite index in fancyL."
This makes it sound as if L is some previously defined object.
How about replacing this by "For L of finite index in fancyL, define..."

p.68, line 10: change "act" to "acts"
A: thanks.

p.68, middle: "...is a purely toric optimal quotient..."
It'd be nice to specify that this is "purely toric at p".
A: good.

p.69, end of WARNING: 3 does not make sense, since the group
has not been identified with Z/42Z.
Anyway it's probably safe to leave this out,
since people reading this will presumably know
what an order 14 subgroup of a cyclic group of order 42 looks like.

p.69, line -3: remove ( to the right of the rightarrow
A: got it.

p.70, conj 4.18: I don't see how #A_i(Q) = #Phi_{A_i} could possibly hold,
given that the former can be infinite, for instance when p=37.
A: I mean "A_i(Q)_tor".

p.72, Table 4.3: where's 67 ???
A: it's omitted.
S: add it?

OK, I'm done (except for section 3.11 on which you wrote
"This section has been rewritten").

I'm done too.