CoCalc Public Fileswww / papers / thesis / poonen-comments.txt
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1Subject:
2        chapters 1,2 of your thesis
3   Date:
4        Sun, 26 Mar 2000 11:53:30 -0800 (PST)
5  From:
6        Bjorn Poonen <[email protected]>
7     To:
8        [email protected]
9
10
11
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
26
27
28p.2, first sentence: if I personally were asked to name the main
29outstanding problem in the arithmetic of elliptic curves, I would say
30it is the problem of whether there is an algorithm to compute
31Mordell-Weil ranks (or equivalently, via descent, the problem of determining
32whether a genus 1 curve over a global field has a rational point).
33Of course this is related to BSD, and in particular is implied by
34the finiteness of Sha, but to me the latter problems are secondary.
35
36p.2, Def 1.1: do you want to require that A be simple over Q?
37(It's up to you.)
38
39p.3, line 7: longterm should be long-term
40
41p.3, Cor 1.4: you could replace "is an integer, up to a unit in"
42                by $\in$
43
44p.4, line -3 (i.e., 3 lines from the bottom): "1-dimensional abelian
45varieties": why not call them elliptic curves
46
47p.5, Thm 1.7: define $\rho_{E,p}$
48
49p.5, line -3: "one expects..."  Is there some theoretical heuristic
50for this?  If not, it might be more accurate to write
51"numerical experiments suggest..."
52Also (to be picky), when you write "most of III(A-dual)"
53you don't really mean most of III(A-dual) for each A,
54but most as you VARY A, I am guessing.
55
56p.6: "So far there is absolutely no evidence..."
57I guess there is no evidence to lead one to conjecture the opposite, either.
58I guess I don't understand your reasons for writing this sentence.
59By writing it this way, do you mean to suggest that you are more
60inclined to believe that III(A-dual) is eventually all visible
61in some J_0(NM)?
62
63p.6, next paragraph: significant difficult
64
65p.6, first sentence of 1.1.6: "...is bound to fail."
66This sounds as if you've proved that it will fail.
67If you haven't, maybe it would be better to say "will probably fail".
68
69p.6, 1.1.6: give a reference for Kani's conjecture.
70
71p.6, first sentence of 1.2: remove the comma
72
73p.6, second sentence of 1.2: "to provably compute"
74(grammatically speaking, it's incorrect to split an infinitive)
75
76p.6, Theorem 1.8: The sentence beginning "Suppose p is an odd prime..."
77sounds a little funny to my ear.  I'd suggest replacing the "and" by
78"or the order"
79
80p.6, last "sentence" of Theorem 1.8:  It's better to avoid using
81a symbol as the verb of a sentence.  You could instead say
82"Then there exists an injection
83        B(Q)/p...."
84
85p.7, a little over halfway down the page:
86I'd suggest putting a period after "K_1=0" and then beginning a new sentence.
87
88p.7, next paragraph: "the latter group contains infinitely many elements
89of order p"   Maybe give a reference for this? (even though you don't
90use it)
91
92p.7, next paragraph: archimedean
93
94p.7, next paragraph: define Q_v^ur.  Also, give a reference for the
95generalization of Tate uniformization.
96
97p.7, same paragraph: "it follows that there is a point Q..."
98Am I missing something?
99It seems to me that this doesn't work when v=p.
100I'm worried...
101
102p.8: when you take stalks the J suddenly becomes a B! (twice)
103
104p.8, middle: need a period at the end of the paragraph
105
106p.8, next paragaph: "The 2-primary subgroup $\Phi$ of $A \cap B$
107is rational over $\Q$."
108I don't see why the points in $\Phi$ have to be rational.
109Oh, do you mean simply that it is rational as a subgroup?
110
111p.8, penultimate sentence of the proof: "the component group...has order
112a power of 2".  In fact, it's trivial, since A-tilde has good reduction at 2.
113
114p.9, line 6: quotient needs an s
115
116p.9, two paragraphs later: "By definition, there must be other subvarieties..."
117By definition of what?
118
119p.9, end of that paragraph: "can not" should be "cannot" I think.
120
121p.10, section title of 1.3.1: move "only" after "considering"
122(only should be put as close as possible to the thing it is onlifying,
123if you know what I mean)
124
125p.15, section 2.1: Perhaps explain the motivation for these
126definitions.  (You probably have more intuition and knowledge about
127this than I do.  Is it that {a,b} was originally thought of as the
128homotopy type of a path from a to b through the upper half plane (or
129its projection in a modular curve).  This would explain the relations,
130for instance.)
131
132p.16, line 2: "torsion-free quotient":  Are you claiming that this
133quotient is already torsion-free, or that you are going to make it
134torsion-free by dividing out its torsion subgroup if necessary?
135If the latter, I think it'd be worth defining the term
136"torsion-free quotient" separately.
137
138p.16, line 2: You never defined Z[epsilon].
139Is it the subring of the space of functions from (Z/NZ)* to C
140generated by epsilon, or the subring of C generated by the values
141of epsilon, or perhaps the group ring Z[G] where G is the group
142generated by epsilon?  (I'm pretty sure I know the answer, but
144
145p.16, definition of M_k(N,epsilon;R): I think you mean
146"tensor over Z[epsilon]".  "Tensor" by itself means "tensor over Z,"
147which will give something very different.
148
149p.17, first two sentences of 2.4.1: This is a little vague (and awkward).
151Let $V$ denote either a space $M_k(N,\epsilon;R)$ of modular symbols
152or a space \dots of modular forms [you should clarify what sort of
153spaces of forms you will consider].
154The Hecke algebra $\T$ is then the subring of $\End_R(V)$
155generated by the $T_n$.
156
157Clearly T depends on the choice of N,epsilon,R.
158But given this data, is it the same for modular symbols
159and for modular forms?  I suppose the answer might depend on
160exactly which type of modular forms you consider.
161
162Is it obvious what the action on antiholomorphic forms is?
163
164p.17, Prop 2.7: Give a reference for this, if you're not going to prove it.
165
166p.18, line 8: it's should be its
167
168p.18, Definition 2.10: since "plus one" is acting as an adjective,
169I think it'd be better to put a hyphen in the middle.
170Same for minus-one.
171
172p.20, Definition 2.14: I have some questions for you: are the new and old
173modular symbols disjoint?  Is their sum equal to the whole space,
174or at least is their sum of finite index in the whole space?
175
176p.20, Remark 2.15: "can not" should be one word I think.
177
178p.20, Remark 2.16: Is p prime to MN?  What is F_p[epsilon]?
179Is it Z[epsilon]/(p), or Z[epsilon]/(fancyp) where fancyp
180is a prime of Z[epsilon] above p, or ... ?
181
182p.20, line 4 of Remark 2.16: basis should be bases
183
184p.20, matrix in Remark 2.16:
185Are you sure you want to write it in this transposed way?
186It is much more common to write linear transformations
187as matrices acting on the left on column vectors.
188(If you are going to keep it as is, it might help to remark
189that you are doing things this way.)
190
191p.20, bottom: It'd be better to define P^1(t) in a separate sentence.
192When I first read this, I didn't realize that this was supposed to
193be a definition of P^1(t) and I started looking back at earlier
194pages searching for one.
195
196p.21, top: In some sense, deterministic algorithms have a greater right
197to be called algorithms than random algorithms.  Although I am sure
198that from the implementation point of view it was easier to do things
199the way you did them, you might at least add a comment that it is
200possible to rewrite this a deterministic algorithm, say by first computing
201coset representatives for Gamma(MN) in Gamma(1),...
202
203p.21, 2.5.1: "base field"?  There has been no mention of base field
204up to now, in the context of modular symbols.
205Do you mean that you are now taking R to be a field?
206By "degeneracy maps" do you mean alpha_t and beta_t
207relativized to R?
208
209p.22, first line of proof of Theorem 2.19: tensor over Z[epsilon] again?
210
211p.26, middle: exists should be exist
212
213p.27, last line of proof of Prop 2.28: "torsion free" should be torsion-free
214
215
216Subject:
218   Date:
219        Mon, 27 Mar 2000 23:27:05 -0800 (PST)
220  From:
221        Bjorn Poonen <[email protected]>
222     To:
223        [email protected]
224
225
226
227
228Dear William:
229
231Below are the rest of my comments.
232
233--Bjorn
234
235
236
237p.28, diagram 2.1, etc.:  I'm really puzzled by this and your
238comment on p.44 that the degree of the composition
239        theta_f : A_f-wedge --> A_f
240need not be a square.  There's no contradiction, but there's
241a natural approach to try to prove that it IS a square,
242and I'm wondering where it goes wrong.  So here are some questions
2441) For k>2 is J_k(N,epsilon) an abelian variety?
245(It was unclear to me from your remark about Shimura at the beginning
246of section 2.7 whether Shimura proved this in general or not.)
2472) If so, is it a PPAV ? I think this is equivalent
248to the complex torus being isomorphic to its dual.
2493) If a complex torus is a quotient of an abelian variety over C,
250is it automatically an abelian variety?  (I think yes.)
2514) Is A_f-wedge --> J_k(N,epsilon) the map dual to J_k(N,epsilon) --> A_f ?
2525) Is theta_f always an isogeny?
253
254
255p.30, bottom: what does it mean to compute an O-module.
256I guess what I'm really asking is, how will you present the answer?
257Will you give a Z-basis?
258
259p.31, 2nd paragraph of 3.2: In the definition of M_k(Gamma) are you
260working over C?
261
262p.32, line 5: "Put R=F_p in Prop 3.6"  -- but just before Prop 3.6
263you said that R was going to be a subring of C.
264
265p.32, paragraph following Lemma 3.11: a_i is an element of what?
266the positive integers?
267
268p.32, end of this paragraph: "We thus represent epsilon as a matrix"
269Why call it a matrix, if it's really just a vector?
270
271p.32, bottom: the ' in n'th looks a lot like an apostrophe here.
272You might try $(n')^{\text{th}}$.
273I personally prefer $n^{\text{th}}$ to $n$th (so much so, that I made
274a macro out of it).  If \text doesn't work in your brand of tex,
275try \operatorname in its place.
276
277p.33, line before definition 3.13:
278I don't understand the (2^{n-2}-1)/2.
279Shouldn't it be 2^{n-3}, for n>=3 ?
280
281p.33, sums in Theorem 3.14: the size and spacing of the indices of
282summation looks really weird.
283
284p.34: delete comma after "cumbersome"
285
286p.34, two lines later: "The author..."  of this thesis or of [Hij64]?
287
288p.34, same sentence: "...has done this and found..."  The tenses don't
289match.  How about "has done this and has found..."
290
291p.34, line -4: what is S?
292
293p.35, line 2 of 3.6.1: something's messed up
294
295p.35, (3.1): this is a little weird in that M_k(N,epsilon)
296is not a K-vector space
297
298p.35, two lines later: how do view the elements of T
299as "sitting inside M_k(N,epsilon)"?
300
301p.35, prop 3.15: Probably you should go back to Def 2.14
302and do it over other bases, since I think here you want new
303modular symbols over K, in order to get a good notion of irreducible.
304
305p.35, next paragraph: "The new and old subspace of M_k(N,epsilon)^perp
306are defined as in Definition 2.14."
307Will the alpha_t and beta_t be replaced by beta_t^perp and alpha_t^perp,
308respectively?
309If so, it might be worth giving the definition in full here
310rather than refer back to Def 2.14.
311
312p.35, algorithm 3.16, lines 4-5: "Using the Hecke operators..."
313Although there's nothing technically wrong with this sentence,
314it tricked me into thinking it was going to be parsed differently,
315if you know what I mean.  Is there some way you could rewrite it?
316
317p.35, algorithm 3.16, 3(b): is this stated correctly?
318Give a reference for the facts you are assuming, or prove them.
319
320p.36, top: "repeat step 1"; do you mean just step 1, or do you
321mean go back to step 1.  also, did you mean to replace p by
322the next larger prime?
323
324All over the place: Some editors consider contractions (like don't)
325too informal for published math.
326
327p.37, alg 3.19: "Then for any randomly chosen..."
328What is the mathematical meaning you have in mind here?
329By the way, is K infinite?
330
331p.37, same sentence: by my convention, g(A)v is always an eigenvector;
332the real question is whether it is nonzero!
333
334p.38, alg 3.20, step 2: by Hecke operator, do you mean a T_n,
335or any linear combination?  If the former, it's not obvious that
336the primitive element theorem is enough.
337
338p.38, step 4: "w is a freely generating Manin symbols".
339Even without the "s" I'm not sure what this means.
340
341p.38, line -5: define K[f].
342
343p.38, bottom: Is it clear that these traces determine f uniquely?
344
345p.39, end of 3.6: is it clear that all ties will eventually be broken?
346
347p.41, just before def 3.25: "...we use it to computing..."
348
349p.42, line 5: "The rank of a square matrix equals the rank of its transpose..."
350This holds even if the matrix is not square!
351
352p.42, first line of proof of 3.29: define O-lattice.
353In particular, make clear that you insist on finite covolume.
354
355p.42, sentence above alg 3.30: J(Q) has not been defined.
356Do you mean to say that when k=2, epsilon=1,
357then J can be identified with J_0(N)(C)?
358
359p.43, 3.9.1: Do you know about Glenn Steven's book,
360called "Arithmetic on modular curves" or something like that?
361I think maybe he works out in general over which fields
362cusps on modular curves are defined.  This together with
363modular symbol calculations should give a reasonable solution
364to the problem.  I'm not saying that you should carry this out;
365but if you feel that Steven's book is relevant, maybe you could
366cite it.
367
368p.43, proof of prop 3.32:
369In what space are T_p and Frob+Ver equal?
370Define g.
371Does all this work even in the bad reduction case?
372Give reference for f(t)=x^{-g} F(x), or explain.
373
374p.43, bottom: Are you claiming that you have a counterexample
375in the form A_f ?
376(By the way, you should use ; or : or . instead of ,
377in the middle of this sentence.)
378
379p.44: give a reference for Prop 3.35
380
381p.44, alg 3.36: define "modular kernel"
382
383p.46: is it known that c_A is a positive integer?
384
385p.47, second line of proof of 3.4: "of" after "smooth locus"
386
387p.47, (3.2): give a reference for the isomorphism in the middle
388
389p.47, middle: should Tor^1 be Tor_1 ?
390
391p.47: "torsion free" should be hyphenated I think (several times)
392
393p.47, line -7: Is fancyB a Neron model too?
394
395p.47, same line: "In particular,..."  How does this follow from
396the exact sequence from Mazur?
397
398p.47, next line: why is the map on the right an isomorphism?
399
400p.48, line 6: singe (I don't think that's the word you want!)
401
402p.48, remark 3.44: peak (wrong word, again)
403
404p.49, top: you define g but never use it!
405
406p.52, middle: "It would be interesting to know whether..."
407Since you then give a counterexample, maybe it would be better
408to replace "whether" by "under what circumstances".
409Also, instead of saying "When k is odd this is clearly not the case"
410it would be better to say "This sometimes fails for odd k"
411since for some odd k and certain N it will be true (for instance
412when S_k(N,epsilon) is trivial!)
413
414p.52, line 2 of 3.13.3: "Section [AL70]"  Is this a typo?
415
416p.52, next line: missing >
417
418p.53: k-2th: put k-2 in parentheses
419
420p.54, top: "to efficiently compute" split infinitive
421
422p.54, line -9: how does e_i depend on i?
423
424p.55, def 3.50: "time" should be "times"
425
426p.55, def 3.50: you shouldn't call it a -1 eigenspace,
427since A_f(C) is not a vector space
428
429p.58, CM elliptic curves: "Let be a rational newform with complex
430multiplication."  What does this mean?  Give a definition or a reference.
431
432p.60, line -6: , after "purely toric" should be .
433
434p.60, line -4: is A' the dual of A (which you later call A-wedge)?
435
436p.61, middle: on the right of the "dualize" should C be C-dual?
437
438p.62, middle: the T and U are backwards in the vertical sequence.
439Think of the semidirect product of G_m by G_a  (the "ax+b" group).
440
441p.62, two lines later: remove the , after "purely toric reduction"
442
443p.62, definition of X_A: should take Homs over the algebraic closure,
444or else define X_A as a group scheme.
445(For example, if T is a nontrivial twist of G_m, then Hom(T,G_m)=0,
446which is not what you want.)
447
448p.62, sequence just before 4.3: give a reference
449
450p.62, thm 4.2: define universal covering
451
452p.64, middle: one-motif !!!
453
454p.65, example 4.7: "... is a Tate curve" over Q_p^ur.
455(For a ramified extension, the answers will be different.)
456
457p.65, middle: define pi_*, pi^*, theta_*, theta^*
458
459p.65, bottom: prove or give a reference for the middle equality
460
461p.66, middle: "Suppose L is of finite index in fancyL."
462This makes it sound as if L is some previously defined object.
463How about replacing this by "For L of finite index in fancyL, define..."
464
465p.68, line 10: change "act" to "acts"
466
467p.68, middle: "...is a purely toric optimal quotient..."
468It'd be nice to specify that this is "purely toric at p".
469
470p.69, end of WARNING: 3 does not make sense, since the group
471has not been identified with Z/42Z.
472Anyway it's probably safe to leave this out,
473since people reading this will presumably know
474what an order 14 subgroup of a cyclic group of order 42 looks like.
475
476p.69, line -3: remove ( to the right of the rightarrow
477
478p.70, conj 4.18: I don't see how #A_i(Q) = #Phi_{A_i} could possibly hold,
479given that the former can be infinite, for instance when p=37.
480
481p.72, Table 4.3: where's 67 ???
482
483
484
485OK, I'm done (except for section 3.11 on which you wrote
486"This section has been rewritten").
487
488
489
490
491