Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1These are Bjorn's comments on my thesis.
3  "A:" or "S:"
5 "S:" means I'm stumped at present (possibly because of lack of net access)
6
7P: is a pointer to where I'm at.
8
9Date: Sun, 26 Mar 2000 11:53:30 -0800 (PST)
10To: [email protected]
11Subject: chapters 1,2 of your thesis
12
13Dear William:
14
15So far I've read through the end of Chapter 2 in your thesis.
16It's really very well written.  I must say, however, that the
17technical nature of Chapter 2 made me want to skim through it
18rather than read every detail; I suppose that's inevitable.
19
20Anyway, here are the comments I have so far.  You can choose to
21ignore most of them if you want; there are very few that are substantial.
22I hope you won't be offended if I sometimes complain about grammar!
23
24--Bjorn
25
26p.2, first sentence: if I personally were asked to name the main
27outstanding problem in the arithmetic of elliptic curves, I would say
28it is the problem of whether there is an algorithm to compute
29Mordell-Weil ranks (or equivalently, via descent, the problem of determining
30whether a genus 1 curve over a global field has a rational point).
31Of course this is related to BSD, and in particular is implied by
32the finiteness of Sha, but to me the latter problems are secondary.
33
34A: I changed the wording slightly, and added references to two papers
35that agree with my opinion.
36
37p.2, Def 1.1: do you want to require that A be simple over Q? (It's
38up to you.)
39A: No; I'll say "simple" when needed.
40
41p.3, line 7: longterm should be long-term
42A: OK.
43
44p.3, Cor 1.4: you could replace "is an integer, up to a unit in"
45		by $\in$
46A: OK.
47
48p.4, line -3 (i.e., 3 lines from the bottom): "1-dimensional abelian
49varieties": why not call them elliptic curves
50A: OK.
51
52p.5, Thm 1.7: define $\rho_{E,p}$
53A: OK.
54
55p.5, line -3: "one expects..."  Is there some theoretical heuristic
56for this?
57A: "Mazur told me so", but didn't really explain it sufficiently.
58He said something about the modular degree annihilating the
59"symmetric square", and the symmetric square should have nothing
60a priori, to do with Sha.
61
62   If not, it might be more accurate to write
63   "numerical experiments suggest..."
64   Also (to be picky), when you write "most of III(A-dual)"
65  you don't really mean most of III(A-dual) for each A,
66  but most as you VARY A, I am guessing.
67A: Right -- thanks.  I changed it to say that
68   "Numerical experiment suggests that
69     as $\Adual$ varies, Sha is often not visible inside
70    J_0(N).  For example ..."
71
72p.6: "So far there is absolutely no evidence..."
73I guess there is no evidence to lead one to conjecture the opposite, either.
74I guess I don't understand your reasons for writing this sentence.
75By writing it this way, do you mean to suggest that you are more
76inclined to believe that III(A-dual) is eventually all visible
77in some J_0(NM)?
78A: I re-worded it to say:
79 "We have been unable to compute any examples in which $\Sha(\Adual)$ is
80  not visible at level~$N$, but becomes visible at some level $NM$.
81  Any data along these lines would be very interesting."
82
83p.6, next paragraph: significant difficult
84A: I must have deleted this or fixed it, as I can't find it.
85
86p.6, first sentence of 1.1.6: "...is bound to fail."
87This sounds as if you've proved that it will fail.
88If you haven't, maybe it would be better to say "will probably fail".
89A: Thanks.
90
91p.6, 1.1.6: give a reference for Kani's conjecture.
93--> See page 9 of Cremona-Mazur: "Kani-Shantz".
94
95p.6, first sentence of 1.2: remove the comma
96A: ok.
97
98p.6, second sentence of 1.2: "to provably compute"
99(grammatically speaking, it's incorrect to split an infinitive)
100A: ok. Now the paragraph reads:
101  Without relying on any unverified conjectures,
102  we will use the following theorem to exhibit abelian varieties~$A$
103  such that the visible part of $\Sha(A)$ is nonzero.
104
105p.6, Theorem 1.8: The sentence beginning "Suppose p is an odd prime..."
106sounds a little funny to my ear.  I'd suggest replacing the "and" by
107"or the order"
108A: done.
109
110p.6, last "sentence" of Theorem 1.8:  It's better to avoid using
111a symbol as the verb of a sentence.  You could instead say
112"Then there exists an injection
113	B(Q)/p...."
114A: good point.
115
116p.7, a little over halfway down the page:
117I'd suggest putting a period after "K_1=0" and then beginning a new sentence.
118A: ok.
119
120p.7, next paragraph: "the latter group contains infinitely many elements
121of order p"   Maybe give a reference for this? (even though you don't
122use it)
123A: ok, I'll mention Shafarevich's paper.
124
125p.7, next paragraph: archimedean
126A: ok
127
128p.7, next paragraph: define Q_v^ur.  Also, give a reference for the
129generalization of Tate uniformization.
130S: look at what Ken does... Faltings-Chai??
131
132p.7, same paragraph: "it follows that there is a point Q..."
133Am I missing something?
134It seems to me that this doesn't work when v=p.
135I'm worried...
136A: It doesn't; I've weekend the statement of the theorem accordingly.
137I wonder how to generalize the result to include this case...?
138
139p.8: when you take stalks the J suddenly becomes a B! (twice)
140A: woops!
141
142p.8, middle: need a period at the end of the paragraph
143A: ok
144
145p.8, next paragaph: "The 2-primary subgroup $\Phi$ of $A \cap B$
146is rational over $\Q$."
147I don't see why the points in $\Phi$ have to be rational.
148Oh, do you mean simply that it is rational as a subgroup?
149A: I mean "rational as a subgroup", which I'll write more explicitly.
150
151p.8, penultimate sentence of the proof: "the component group...has order
152a power of 2".  In fact, it's trivial, since A-tilde has good reduction at 2.
153A: Thanks for pointing this out.
154
155p.9, line 6: quotient needs an s
156A: Thanks; also, I just realized that "rank zero optimal quotients"
157should be "rank-zero optimal quotients".
158
159p.9, two paragraphs later: "By definition, there must be other subvarieties..."
160By definition of what?
161A: modular degree -- I reworded this paragraph more clearly.
162
163p.9, end of that paragraph: "can not" should be "cannot" I think.
164A: I agree.
165
166p.10, section title of 1.3.1: move "only" after "considering"
167(only should be put as close as possible to the thing it is onlifying,
168if you know what I mean)
169A: Thanks for the tip!
170
171p.15, section 2.1: Perhaps explain the motivation for these
172definitions.  (You probably have more intuition and knowledge about
173this than I do.  Is it that {a,b} was originally thought of as the
174homotopy type of a path from a to b through the upper half plane (or
175its projection in a modular curve).  This would explain the relations,
176for instance.)
178
179p.16, line 2: "torsion-free quotient":  Are you claiming that this
180quotient is already torsion-free, or that you are going to make it
181torsion-free by dividing out its torsion subgroup if necessary?
182If the latter, I think it'd be worth defining the term
183"torsion-free quotient" separately.
184A: I've systematically written "the largest torsion-free quotient of" in place
185   of "torsion-free quotient."
186
187p.16, line 2: You never defined Z[epsilon].
188Is it the subring of the space of functions from (Z/NZ)* to C
189generated by epsilon, or the subring of C generated by the values
190of epsilon, or perhaps the group ring Z[G] where G is the group
191generated by epsilon?  (I'm pretty sure I know the answer, but
193A:  Thanks for pointing out the ambiguity.
194It's the subring of C generated by the values of eps.
195
196p.16, definition of M_k(N,epsilon;R): I think you mean
197"tensor over Z[epsilon]".  "Tensor" by itself means "tensor over Z,"
198which will give something very different.
199A: Yep. Thanks.
200
201p.17, first two sentences of 2.4.1: This is a little vague (and awkward).
203Let $V$ denote either a space $M_k(N,\epsilon;R)$ of modular symbols
204or a space \dots of modular forms [you should clarify what sort of
205spaces of forms you will consider].
206The Hecke algebra $\T$ is then the subring of $\End_R(V)$
207generated by the $T_n$.
208
209Clearly T depends on the choice of N,epsilon,R.
210But given this data, is it the same for modular symbols
211and for modular forms?  I suppose the answer might depend on
212exactly which type of modular forms you consider.
213
214Is it obvious what the action on antiholomorphic forms is?
215A: Ok.
216
217p.17, Prop 2.7: Give a reference for this, if you're not going to prove it.
218A: Ok.
219
220p.18, line 8: it's should be its
221A: Ok.
222
223p.18, Definition 2.10: since "plus one" is acting as an adjective,
224I think it'd be better to put a hyphen in the middle.
225Same for minus-one.
226A: Ok.
227
228p.20, Definition 2.14: I have some questions for you: are the new and old
229modular symbols disjoint?  Is their sum equal to the whole space,
230or at least is their sum of finite index in the whole space?
231S: I'm not sure about the sum on the Eisenstein part
232
233p.20, Remark 2.15: "can not" should be one word I think.
234
235p.20, Remark 2.16: Is p prime to MN?  What is F_p[epsilon]?
236Is it Z[epsilon]/(p), or Z[epsilon]/(fancyp) where fancyp
237is a prime of Z[epsilon] above p, or ... ?
238
239p.20, line 4 of Remark 2.16: basis should be bases
240
241p.20, matrix in Remark 2.16:
242Are you sure you want to write it in this transposed way?
243It is much more common to write linear transformations
244as matrices acting on the left on column vectors.
245(If you are going to keep it as is, it might help to remark
246that you are doing things this way.)
247
248p.20, bottom: It'd be better to define P^1(t) in a separate sentence.
249When I first read this, I didn't realize that this was supposed to
250be a definition of P^1(t) and I started looking back at earlier
251pages searching for one.
252
253p.21, top: In some sense, deterministic algorithms have a greater right
254to be called algorithms than random algorithms.  Although I am sure
255that from the implementation point of view it was easier to do things
256the way you did them, you might at least add a comment that it is
257possible to rewrite this a deterministic algorithm, say by first computing
258coset representatives for Gamma(MN) in Gamma(1),...
259
260p.21, 2.5.1: "base field"?  There has been no mention of base field
261up to now, in the context of modular symbols.
262Do you mean that you are now taking R to be a field?
263By "degeneracy maps" do you mean alpha_t and beta_t
264relativized to R?
265
266p.22, first line of proof of Theorem 2.19: tensor over Z[epsilon] again?
267
268p.26, middle: exists should be exist
269
270p.27, last line of proof of Prop 2.28: "torsion free" should be torsion-free
271
272
273Date: Mon, 27 Mar 2000 23:27:05 -0800 (PST)
274To: [email protected]
276
277Dear William:
278
280Below are the rest of my comments.
281
282--Bjorn
283
284 p.28, diagram 2.1, etc.:  I'm really puzzled by this and your
285 comment on p.44 that the degree of the composition
286 	theta_f : A_f-wedge --> A_f
287 need not be a square.  There's no contradiction, but there's
288 a natural approach to try to prove that it IS a square,
289 and I'm wondering where it goes wrong.  So here are some questions
291 1) For k>2 is J_k(N,epsilon) an abelian variety?
292 (It was unclear to me from your remark about Shimura at the beginning
293 of section 2.7 whether Shimura proved this in general or not.)
294
295A: Shimura's construction is different; there's no reason why ours should agree.
296
297 2) If so, is it a PPAV ? I think this is equivalent
298 to the complex torus being isomorphic to its dual.
299
300A: I think so, by properties of the Peterson inner product.
301
302 3) If a complex torus is a quotient of an abelian variety over C,
303 is it automatically an abelian variety?  (I think yes.)
304
305A: Yes, at least in this case, because it's a quotient by an ideal of
306endomorphisms.
307
308 4) Is A_f-wedge --> J_k(N,epsilon) the map dual to J_k(N,epsilon) --> A_f ?
309
310A: It should be, in the Shimura setting.
311
312 5) Is theta_f always an isogeny?
313
314A: I think so...
315   Maybe your above remark can be used to show that my construction is definitely
316   {\em not} Shimura's.
317
318p.30, bottom: what does it mean to compute an O-module.
319I guess what I'm really asking is, how will you present the answer?
320Will you give a Z-basis?
321A: I don't really care, because I'm not analyzing efficiency.
322The algorithm here takes "compute O-module" as a black box;
323I think it would be a mistake for me to by precise about how this part of the
324computation should go...  (I've added a paranthetical remark of this nature.)
325
326p.31, 2nd paragraph of 3.2: In the definition of M_k(Gamma) are you
327working over C?
328A: fixed.
329p.32, line 5: "Put R=F_p in Prop 3.6"  -- but just before Prop 3.6
330you said that R was going to be a subring of C.
331A: fixed.
332
333p.32, paragraph following Lemma 3.11: a_i is an element of what?
334the positive integers?
335A: no, of the group it generates.
336
337p.32, end of this paragraph: "We thus represent epsilon as a matrix"
338Why call it a matrix, if it's really just a vector?
339A: woops -- thanks
340
341p.32, bottom: the ' in n'th looks a lot like an apostrophe here.
342You might try $(n')^{\text{th}}$.
343I personally prefer $n^{\text{th}}$ to $n$th (so much so, that I made
344a macro out of it).  If \text doesn't work in your brand of tex,
345try \operatorname in its place.
346A: Some style manuals explicitly told me not to write n^{th}... and I agree with their argument.
347I'll use paranthesis.
348
349p.33, line before definition 3.13:
350I don't understand the (2^{n-2}-1)/2.
351Shouldn't it be 2^{n-3}, for n>=3 ?
352A: Thanks!!!
353
354p.33, sums in Theorem 3.14: the size and spacing of the indices of
355summation looks really weird.
356A: OK; I changed to summing over x in (Z/NZ)^*, which is more precise and eliminates
357the need for all of the "mod"'s.
358
359p.34: delete comma after "cumbersome"
360A: I disagree.  It is an independent clause, so it is set off in commas.
361I could delete "more cumbersome" and the sentence still makes sense.
362If I had said "Alternatively, a more cumbersome, way to ..." then I should have
363deleted the comma.
364
365p.34, two lines later: "The author..."  of this thesis or of [Hij64]?
366A: "of this thesis". fixed.
367
368p.34, same sentence: "...has done this and found..."  The tenses don't
369match.  How about "has done this and has found..."
370A: I did it.
371
372p.34, line -4: what is S?
373A: Oops -- a space of cusp forms.
374
375p.35, line 2 of 3.6.1: something's messed up
376A: Oops -- there was an extra "\".
377
378p.35, (3.1): this is a little weird in that M_k(N,epsilon)
379is not a K-vector space
380A: changed to ";K"
381
382p.35, two lines later: how do view the elements of T
383as "sitting inside M_k(N,epsilon)"?
384A: I mean, as endomorphisms of ...
385
386p.35, prop 3.15: Probably you should go back to Def 2.14
387and do it over other bases, since I think here you want new
388modular symbols over K, in order to get a good notion of irreducible.
389A: ok.
390
391p.35, next paragraph: "The new and old subspace of M_k(N,epsilon)^perp
392are defined as in Definition 2.14."
393Will the alpha_t and beta_t be replaced by beta_t^perp and alpha_t^perp,
394respectively?
395If so, it might be worth giving the definition in full here
396rather than refer back to Def 2.14.
397
398p.35, algorithm 3.16, lines 4-5: "Using the Hecke operators..."
399Although there's nothing technically wrong with this sentence,
400it tricked me into thinking it was going to be parsed differently,
401if you know what I mean.  Is there some way you could rewrite it?
402A: re-ordered
403
404P: <------------------>
405
406p.35, algorithm 3.16, 3(b): is this stated correctly?
407Give a reference for the facts you are assuming, or prove them.
408S: OK.  This was incorrectly and awkwardly stated. I've fixed it.
409(But need to add a real reference; perhaps to Loic's paper.)
410
411p.36, top: "repeat step 1"; do you mean just step 1, or do you
412mean go back to step 1.  also, did you mean to replace p by
413the next larger prime?
414A: Thanks; this was ridiculous imprecise as it stood!
415
416All over the place: Some editors consider contractions (like don't)
417too informal for published math.
418A: I just fixed it with grep.
419
420p.37, alg 3.19: "Then for any randomly chosen..."
421What is the mathematical meaning you have in mind here?
422A: As you point out, there's no need to have the word "random".
423By the way, is K infinite?
424A: NOPE.
425
426
427p.37, same sentence: by my convention, g(A)v is always an eigenvector;
428the real question is whether it is nonzero!
429A: Oops!  Your convention is my convention, too!  Thanks.  I fixed it.
430
431
432p.38, alg 3.20, step 2: by Hecke operator, do you mean a T_n,
433or any linear combination?  If the former, it's not obvious that
434the primitive element theorem is enough.
435A: Woops.  I mean the *latter*.  Not only is the former "not obvious", it isn't
436true!  I found the first example in the course of my computations; it
437occurs in S_2(Gamma_0(512)).  I've added a note to this effect.
438
439
440p.38, step 4: "w is a freely generating Manin symbols".
441Even without the "s" I'm not sure what this means.
442A: a symbol of the form [P,(c,d)].
443
444p.38, line -5: define K[f].
445A: done.
446
447p.38, bottom: Is it clear that these traces determine f uniquely?
448A: No, but I'm sure it's true.
449However, it is clear if we include Tr(a_n), for all n.  I've
450changed to this, because it is also cleaner.
451
452p.39, end of 3.6: is it clear that all ties will eventually be broken?
453A: I don't know.
454
455p.41, just before def 3.25: "...we use it to computing..."
456A: fixed.
457
458p.42, line 5: "The rank of a square matrix equals the rank of its transpose..."
459This holds even if the matrix is not square!
460A: Ok!
461
462p.42, first line of proof of 3.29: define O-lattice.
463In particular, make clear that you insist on finite covolume.
464
465p.42, sentence above alg 3.30: J(Q) has not been defined.
466Do you mean to say that when k=2, epsilon=1,
467then J can be identified with J_0(N)(C)?
468A: Yep.
469
470p.43, 3.9.1: Do you know about Glenn Steven's book,
471called "Arithmetic on modular curves" or something like that?
472I think maybe he works out in general over which fields
473cusps on modular curves are defined.  This together with
474modular symbol calculations should give a reasonable solution
475to the problem.  I'm not saying that you should carry this out;
476but if you feel that Steven's book is relevant, maybe you could
477cite it.
478
479A: I'm citing it.  I'll look into this in detail after I finish
480my thesis.
481
482p.43, proof of prop 3.32:
483In what space are T_p and Frob+Ver equal?
484Define g.
485Does all this work even in the bad reduction case?
486Give reference for f(t)=x^{-g} F(x), or explain.
487A: I cleaned this up a lot; strangely enough, I was giving the proof
489S: I still need to give a reference for "f(t)=x^{-g} F(x)".
490
491p.43, bottom: Are you claiming that you have a counterexample
492in the form A_f ?
493(By the way, you should use ; or : or . instead of ,
494in the middle of this sentence.)
495A: Yep!  I should state all this for general abelian varieties
496$A$... and give a reference.
497S: give a reference; see Cassels-Flynn...
498
499p.44: give a reference for Prop 3.35
500A: I don't know one; though, I think it is easy to prove...
501   I never use this proposition, so I'll de-proposition it, and only
502   mention it in passing.
503
504p.44, alg 3.36: define "modular kernel"
505A: done.
506
507p.46: is it known that c_A is a positive integer?
508A: "YES.
509
510p.47, second line of proof of 3.4: "of" after "smooth locus"
511A: fixed, with ou.
512
513p.47, (3.2): give a reference for the isomorphism in the middle
514A: done.
515
516p.47, middle: should Tor^1 be Tor_1 ?
517A: done.
518
519p.47: "torsion free" should be hyphenated I think (several times)
520A: hmm.
521
522p.47, line -7: Is fancyB a Neron model too?
523...
524
525p.47, same line: "In particular,..."  How does this follow from
526the exact sequence from Mazur?
527...
528
529p.47, next line: why is the map on the right an isomorphism?
530...
531
532p.48, line 6: singe (I don't think that's the word you want!)
533...
534
535p.48, remark 3.44: peak (wrong word, again)
536
537p.49, top: you define g but never use it!
538A: I mean "f".
539
540p.52, middle: "It would be interesting to know whether..."
541Since you then give a counterexample, maybe it would be better
542to replace "whether" by "under what circumstances".
543Also, instead of saying "When k is odd this is clearly not the case"
544it would be better to say "This sometimes fails for odd k"
545since for some odd k and certain N it will be true (for instance
546when S_k(N,epsilon) is trivial!)
547A: thanks!!
548
549p.52, line 2 of 3.13.3: "Section [AL70]"  Is this a typo?
550A: oops; I used "cite" instead of "ref".
551
552p.52, next line: missing >
553A: got it.
554
555p.53: k-2th: put k-2 in parentheses
556A: Ok.
557
558p.54, top: "to efficiently compute" split infinitive
559A: *doh*
560
561p.54, line -9: how does e_i depend on i?
562A: woops.
563
564p.55, def 3.50: "time" should be "times"
565A: OK.
566
567p.55, def 3.50: you shouldn't call it a -1 eigenspace,
568since A_f(C) is not a vector space
569A: Ok.
570
571p.58, CM elliptic curves: "Let be a rational newform with complex
572multiplication."  What does this mean?  Give a definition or a reference.
573A: I'll give a silly, but correct, definition.
574
575p.60, line -6: , after "purely toric" should be .
576A:  thanks.
577
578p.60, line -4: is A' the dual of A (which you later call A-wedge)?
579A: yep; you guessed it, I changed my mind at some point...
580
581p.61, middle: on the right of the "dualize" should C be C-dual?
582A: yep.
583
584p.62, middle: the T and U are backwards in the vertical sequence.
585Think of the semidirect product of G_m by G_a  (the "ax+b" group).
586A: I disagree with you here.   The torus is the connected sub-thing,
587not the quotient.
588
589p.62, two lines later: remove the , after "purely toric reduction"
590A: woops.
591
592p.62, definition of X_A: should take Homs over the algebraic closure,
593or else define X_A as a group scheme.
594(For example, if T is a nontrivial twist of G_m, then Hom(T,G_m)=0,
595which is not what you want.)
596A: yep. thanks.
597
598p.62, sequence just before 4.3: give a reference
599A: ok.
600S: but I should be more precise.
601
602p.62, thm 4.2: define universal covering
603A: refered to coleman's paper where everything is defined.
604
605p.64, middle: one-motif !!!
606A: got it.
607
608p.65, example 4.7: "... is a Tate curve" over Q_p^ur.
609(For a ramified extension, the answers will be different.)
610A: ok.
611
612p.65, middle: define pi_*, pi^*, theta_*, theta^*
613A: ok.
614
615p.65, bottom: prove or give a reference for the middle equality
616S: I'll have to dig up a reference for this compatibility when I get back.
617
618p.66, middle: "Suppose L is of finite index in fancyL."
619This makes it sound as if L is some previously defined object.
620How about replacing this by "For L of finite index in fancyL, define..."
621
622p.68, line 10: change "act" to "acts"
623A: thanks.
624
625p.68, middle: "...is a purely toric optimal quotient..."
626It'd be nice to specify that this is "purely toric at p".
627A: good.
628
629p.69, end of WARNING: 3 does not make sense, since the group
630has not been identified with Z/42Z.
631Anyway it's probably safe to leave this out,
632since people reading this will presumably know
633what an order 14 subgroup of a cyclic group of order 42 looks like.
634
635p.69, line -3: remove ( to the right of the rightarrow
636A: got it.
637
638p.70, conj 4.18: I don't see how #A_i(Q) = #Phi_{A_i} could possibly hold,
639given that the former can be infinite, for instance when p=37.
640A: I mean "A_i(Q)_tor".
641
642p.72, Table 4.3: where's 67 ???
643A: it's omitted.