Open with one click!
1
Subject:
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
13
Dear William:
14
15
So far I've read through the end of Chapter 2 in your thesis.
16
It's really very well written. I must say, however, that the
17
technical nature of Chapter 2 made me want to skim through it
18
rather than read every detail; I suppose that's inevitable.
19
20
Anyway, here are the comments I have so far. You can choose to
21
ignore most of them if you want; there are very few that are substantial.
22
I hope you won't be offended if I sometimes complain about grammar!
23
24
--Bjorn
25
26
27
28
p.2, first sentence: if I personally were asked to name the main
29
outstanding problem in the arithmetic of elliptic curves, I would say
30
it is the problem of whether there is an algorithm to compute
31
Mordell-Weil ranks (or equivalently, via descent, the problem of determining
32
whether a genus 1 curve over a global field has a rational point).
33
Of course this is related to BSD, and in particular is implied by
34
the finiteness of Sha, but to me the latter problems are secondary.
35
36
p.2, Def 1.1: do you want to require that A be simple over Q?
37
(It's up to you.)
38
39
p.3, line 7: longterm should be long-term
40
41
p.3, Cor 1.4: you could replace "is an integer, up to a unit in"
42
by $\in$
43
44
p.4, line -3 (i.e., 3 lines from the bottom): "1-dimensional abelian
45
varieties": why not call them elliptic curves
46
47
p.5, Thm 1.7: define $\rho_{E,p}$
48
49
p.5, line -3: "one expects..." Is there some theoretical heuristic
50
for this? If not, it might be more accurate to write
51
"numerical experiments suggest..."
52
Also (to be picky), when you write "most of III(A-dual)"
53
you don't really mean most of III(A-dual) for each A,
54
but most as you VARY A, I am guessing.
55
56
p.6: "So far there is absolutely no evidence..."
57
I guess there is no evidence to lead one to conjecture the opposite, either.
58
I guess I don't understand your reasons for writing this sentence.
59
By writing it this way, do you mean to suggest that you are more
60
inclined to believe that III(A-dual) is eventually all visible
61
in some J_0(NM)?
62
63
p.6, next paragraph: significant difficult
64
65
p.6, first sentence of 1.1.6: "...is bound to fail."
66
This sounds as if you've proved that it will fail.
67
If you haven't, maybe it would be better to say "will probably fail".
68
69
p.6, 1.1.6: give a reference for Kani's conjecture.
70
71
p.6, first sentence of 1.2: remove the comma
72
73
p.6, second sentence of 1.2: "to provably compute"
74
(grammatically speaking, it's incorrect to split an infinitive)
75
76
p.6, Theorem 1.8: The sentence beginning "Suppose p is an odd prime..."
77
sounds a little funny to my ear. I'd suggest replacing the "and" by
78
"or the order"
79
80
p.6, last "sentence" of Theorem 1.8: It's better to avoid using
81
a symbol as the verb of a sentence. You could instead say
82
"Then there exists an injection
83
B(Q)/p...."
84
85
p.7, a little over halfway down the page:
86
I'd suggest putting a period after "K_1=0" and then beginning a new sentence.
87
88
p.7, next paragraph: "the latter group contains infinitely many elements
89
of order p" Maybe give a reference for this? (even though you don't
90
use it)
91
92
p.7, next paragraph: archimedean
93
94
p.7, next paragraph: define Q_v^ur. Also, give a reference for the
95
generalization of Tate uniformization.
96
97
p.7, same paragraph: "it follows that there is a point Q..."
98
Am I missing something?
99
It seems to me that this doesn't work when v=p.
100
I'm worried...
101
102
p.8: when you take stalks the J suddenly becomes a B! (twice)
103
104
p.8, middle: need a period at the end of the paragraph
105
106
p.8, next paragaph: "The 2-primary subgroup $\Phi$ of $A \cap B$
107
is rational over $\Q$."
108
I don't see why the points in $\Phi$ have to be rational.
109
Oh, do you mean simply that it is rational as a subgroup?
110
111
p.8, penultimate sentence of the proof: "the component group...has order
112
a power of 2". In fact, it's trivial, since A-tilde has good reduction at 2.
113
114
p.9, line 6: quotient needs an s
115
116
p.9, two paragraphs later: "By definition, there must be other subvarieties..."
117
By definition of what?
118
119
p.9, end of that paragraph: "can not" should be "cannot" I think.
120
121
p.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,
123
if you know what I mean)
124
125
p.15, section 2.1: Perhaps explain the motivation for these
126
definitions. (You probably have more intuition and knowledge about
127
this than I do. Is it that {a,b} was originally thought of as the
128
homotopy type of a path from a to b through the upper half plane (or
129
its projection in a modular curve). This would explain the relations,
130
for instance.)
131
132
p.16, line 2: "torsion-free quotient": Are you claiming that this
133
quotient is already torsion-free, or that you are going to make it
134
torsion-free by dividing out its torsion subgroup if necessary?
135
If the latter, I think it'd be worth defining the term
136
"torsion-free quotient" separately.
137
138
p.16, line 2: You never defined Z[epsilon].
139
Is it the subring of the space of functions from (Z/NZ)* to C
140
generated by epsilon, or the subring of C generated by the values
141
of epsilon, or perhaps the group ring Z[G] where G is the group
142
generated by epsilon? (I'm pretty sure I know the answer, but
143
another reader might not.)
144
145
p.16, definition of M_k(N,epsilon;R): I think you mean
146
"tensor over Z[epsilon]". "Tensor" by itself means "tensor over Z,"
147
which will give something very different.
148
149
p.17, first two sentences of 2.4.1: This is a little vague (and awkward).
150
Maybe write instead:
151
Let $V$ denote either a space $M_k(N,\epsilon;R)$ of modular symbols
152
or a space \dots of modular forms [you should clarify what sort of
153
spaces of forms you will consider].
154
The Hecke algebra $\T$ is then the subring of $\End_R(V)$
155
generated by the $T_n$.
156
157
Clearly T depends on the choice of N,epsilon,R.
158
But given this data, is it the same for modular symbols
159
and for modular forms? I suppose the answer might depend on
160
exactly which type of modular forms you consider.
161
162
Is it obvious what the action on antiholomorphic forms is?
163
164
p.17, Prop 2.7: Give a reference for this, if you're not going to prove it.
165
166
p.18, line 8: it's should be its
167
168
p.18, Definition 2.10: since "plus one" is acting as an adjective,
169
I think it'd be better to put a hyphen in the middle.
170
Same for minus-one.
171
172
p.20, Definition 2.14: I have some questions for you: are the new and old
173
modular symbols disjoint? Is their sum equal to the whole space,
174
or at least is their sum of finite index in the whole space?
175
176
p.20, Remark 2.15: "can not" should be one word I think.
177
178
p.20, Remark 2.16: Is p prime to MN? What is F_p[epsilon]?
179
Is it Z[epsilon]/(p), or Z[epsilon]/(fancyp) where fancyp
180
is a prime of Z[epsilon] above p, or ... ?
181
182
p.20, line 4 of Remark 2.16: basis should be bases
183
184
p.20, matrix in Remark 2.16:
185
Are you sure you want to write it in this transposed way?
186
It is much more common to write linear transformations
187
as matrices acting on the left on column vectors.
188
(If you are going to keep it as is, it might help to remark
189
that you are doing things this way.)
190
191
p.20, bottom: It'd be better to define P^1(t) in a separate sentence.
192
When I first read this, I didn't realize that this was supposed to
193
be a definition of P^1(t) and I started looking back at earlier
194
pages searching for one.
195
196
p.21, top: In some sense, deterministic algorithms have a greater right
197
to be called algorithms than random algorithms. Although I am sure
198
that from the implementation point of view it was easier to do things
199
the way you did them, you might at least add a comment that it is
200
possible to rewrite this a deterministic algorithm, say by first computing
201
coset representatives for Gamma(MN) in Gamma(1),...
202
203
p.21, 2.5.1: "base field"? There has been no mention of base field
204
up to now, in the context of modular symbols.
205
Do you mean that you are now taking R to be a field?
206
By "degeneracy maps" do you mean alpha_t and beta_t
207
relativized to R?
208
209
p.22, first line of proof of Theorem 2.19: tensor over Z[epsilon] again?
210
211
p.26, middle: exists should be exist
212
213
p.27, last line of proof of Prop 2.28: "torsion free" should be torsion-free
214
215
216
Subject:
217
more comments
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
228
Dear William:
229
230
I've finished "reading" your thesis.
231
Below are the rest of my comments.
232
233
--Bjorn
234
235
236
237
p.28, diagram 2.1, etc.: I'm really puzzled by this and your
238
comment on p.44 that the degree of the composition
239
theta_f : A_f-wedge --> A_f
240
need not be a square. There's no contradiction, but there's
241
a natural approach to try to prove that it IS a square,
242
and I'm wondering where it goes wrong. So here are some questions
243
about the situation:
244
1) 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
246
of section 2.7 whether Shimura proved this in general or not.)
247
2) If so, is it a PPAV ? I think this is equivalent
248
to the complex torus being isomorphic to its dual.
249
3) If a complex torus is a quotient of an abelian variety over C,
250
is it automatically an abelian variety? (I think yes.)
251
4) Is A_f-wedge --> J_k(N,epsilon) the map dual to J_k(N,epsilon) --> A_f ?
252
5) Is theta_f always an isogeny?
253
254
255
p.30, bottom: what does it mean to compute an O-module.
256
I guess what I'm really asking is, how will you present the answer?
257
Will you give a Z-basis?
258
259
p.31, 2nd paragraph of 3.2: In the definition of M_k(Gamma) are you
260
working over C?
261
262
p.32, line 5: "Put R=F_p in Prop 3.6" -- but just before Prop 3.6
263
you said that R was going to be a subring of C.
264
265
p.32, paragraph following Lemma 3.11: a_i is an element of what?
266
the positive integers?
267
268
p.32, end of this paragraph: "We thus represent epsilon as a matrix"
269
Why call it a matrix, if it's really just a vector?
270
271
p.32, bottom: the ' in n'th looks a lot like an apostrophe here.
272
You might try $(n')^{\text{th}}$.
273
I personally prefer $n^{\text{th}}$ to $n$th (so much so, that I made
274
a macro out of it). If \text doesn't work in your brand of tex,
275
try \operatorname in its place.
276
277
p.33, line before definition 3.13:
278
I don't understand the (2^{n-2}-1)/2.
279
Shouldn't it be 2^{n-3}, for n>=3 ?
280
281
p.33, sums in Theorem 3.14: the size and spacing of the indices of
282
summation looks really weird.
283
284
p.34: delete comma after "cumbersome"
285
286
p.34, two lines later: "The author..." of this thesis or of [Hij64]?
287
288
p.34, same sentence: "...has done this and found..." The tenses don't
289
match. How about "has done this and has found..."
290
291
p.34, line -4: what is S?
292
293
p.35, line 2 of 3.6.1: something's messed up
294
295
p.35, (3.1): this is a little weird in that M_k(N,epsilon)
296
is not a K-vector space
297
298
p.35, two lines later: how do view the elements of T
299
as "sitting inside M_k(N,epsilon)"?
300
301
p.35, prop 3.15: Probably you should go back to Def 2.14
302
and do it over other bases, since I think here you want new
303
modular symbols over K, in order to get a good notion of irreducible.
304
305
p.35, next paragraph: "The new and old subspace of M_k(N,epsilon)^perp
306
are defined as in Definition 2.14."
307
Will the alpha_t and beta_t be replaced by beta_t^perp and alpha_t^perp,
308
respectively?
309
If so, it might be worth giving the definition in full here
310
rather than refer back to Def 2.14.
311
312
p.35, algorithm 3.16, lines 4-5: "Using the Hecke operators..."
313
Although there's nothing technically wrong with this sentence,
314
it tricked me into thinking it was going to be parsed differently,
315
if you know what I mean. Is there some way you could rewrite it?
316
317
p.35, algorithm 3.16, 3(b): is this stated correctly?
318
Give a reference for the facts you are assuming, or prove them.
319
320
p.36, top: "repeat step 1"; do you mean just step 1, or do you
321
mean go back to step 1. also, did you mean to replace p by
322
the next larger prime?
323
324
All over the place: Some editors consider contractions (like don't)
325
too informal for published math.
326
327
p.37, alg 3.19: "Then for any randomly chosen..."
328
What is the mathematical meaning you have in mind here?
329
By the way, is K infinite?
330
331
p.37, same sentence: by my convention, g(A)v is always an eigenvector;
332
the real question is whether it is nonzero!
333
334
p.38, alg 3.20, step 2: by Hecke operator, do you mean a T_n,
335
or any linear combination? If the former, it's not obvious that
336
the primitive element theorem is enough.
337
338
p.38, step 4: "w is a freely generating Manin symbols".
339
Even without the "s" I'm not sure what this means.
340
341
p.38, line -5: define K[f].
342
343
p.38, bottom: Is it clear that these traces determine f uniquely?
344
345
p.39, end of 3.6: is it clear that all ties will eventually be broken?
346
347
p.41, just before def 3.25: "...we use it to computing..."
348
349
p.42, line 5: "The rank of a square matrix equals the rank of its transpose..."
350
This holds even if the matrix is not square!
351
352
p.42, first line of proof of 3.29: define O-lattice.
353
In particular, make clear that you insist on finite covolume.
354
355
p.42, sentence above alg 3.30: J(Q) has not been defined.
356
Do you mean to say that when k=2, epsilon=1,
357
then J can be identified with J_0(N)(C)?
358
359
p.43, 3.9.1: Do you know about Glenn Steven's book,
360
called "Arithmetic on modular curves" or something like that?
361
I think maybe he works out in general over which fields
362
cusps on modular curves are defined. This together with
363
modular symbol calculations should give a reasonable solution
364
to the problem. I'm not saying that you should carry this out;
365
but if you feel that Steven's book is relevant, maybe you could
366
cite it.
367
368
p.43, proof of prop 3.32:
369
In what space are T_p and Frob+Ver equal?
370
Define g.
371
Does all this work even in the bad reduction case?
372
Give reference for f(t)=x^{-g} F(x), or explain.
373
374
p.43, bottom: Are you claiming that you have a counterexample
375
in the form A_f ?
376
(By the way, you should use ; or : or . instead of ,
377
in the middle of this sentence.)
378
379
p.44: give a reference for Prop 3.35
380
381
p.44, alg 3.36: define "modular kernel"
382
383
p.46: is it known that c_A is a positive integer?
384
385
p.47, second line of proof of 3.4: "of" after "smooth locus"
386
387
p.47, (3.2): give a reference for the isomorphism in the middle
388
389
p.47, middle: should Tor^1 be Tor_1 ?
390
391
p.47: "torsion free" should be hyphenated I think (several times)
392
393
p.47, line -7: Is fancyB a Neron model too?
394
395
p.47, same line: "In particular,..." How does this follow from
396
the exact sequence from Mazur?
397
398
p.47, next line: why is the map on the right an isomorphism?
399
400
p.48, line 6: singe (I don't think that's the word you want!)
401
402
p.48, remark 3.44: peak (wrong word, again)
403
404
p.49, top: you define g but never use it!
405
406
p.52, middle: "It would be interesting to know whether..."
407
Since you then give a counterexample, maybe it would be better
408
to replace "whether" by "under what circumstances".
409
Also, instead of saying "When k is odd this is clearly not the case"
410
it would be better to say "This sometimes fails for odd k"
411
since for some odd k and certain N it will be true (for instance
412
when S_k(N,epsilon) is trivial!)
413
414
p.52, line 2 of 3.13.3: "Section [AL70]" Is this a typo?
415
416
p.52, next line: missing >
417
418
p.53: k-2th: put k-2 in parentheses
419
420
p.54, top: "to efficiently compute" split infinitive
421
422
p.54, line -9: how does e_i depend on i?
423
424
p.55, def 3.50: "time" should be "times"
425
426
p.55, def 3.50: you shouldn't call it a -1 eigenspace,
427
since A_f(C) is not a vector space
428
429
p.58, CM elliptic curves: "Let be a rational newform with complex
430
multiplication." What does this mean? Give a definition or a reference.
431
432
p.60, line -6: , after "purely toric" should be .
433
434
p.60, line -4: is A' the dual of A (which you later call A-wedge)?
435
436
p.61, middle: on the right of the "dualize" should C be C-dual?
437
438
p.62, middle: the T and U are backwards in the vertical sequence.
439
Think of the semidirect product of G_m by G_a (the "ax+b" group).
440
441
p.62, two lines later: remove the , after "purely toric reduction"
442
443
p.62, definition of X_A: should take Homs over the algebraic closure,
444
or 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,
446
which is not what you want.)
447
448
p.62, sequence just before 4.3: give a reference
449
450
p.62, thm 4.2: define universal covering
451
452
p.64, middle: one-motif !!!
453
454
p.65, example 4.7: "... is a Tate curve" over Q_p^ur.
455
(For a ramified extension, the answers will be different.)
456
457
p.65, middle: define pi_*, pi^*, theta_*, theta^*
458
459
p.65, bottom: prove or give a reference for the middle equality
460
461
p.66, middle: "Suppose L is of finite index in fancyL."
462
This makes it sound as if L is some previously defined object.
463
How about replacing this by "For L of finite index in fancyL, define..."
464
465
p.68, line 10: change "act" to "acts"
466
467
p.68, middle: "...is a purely toric optimal quotient..."
468
It'd be nice to specify that this is "purely toric at p".
469
470
p.69, end of WARNING: 3 does not make sense, since the group
471
has not been identified with Z/42Z.
472
Anyway it's probably safe to leave this out,
473
since people reading this will presumably know
474
what an order 14 subgroup of a cyclic group of order 42 looks like.
475
476
p.69, line -3: remove ( to the right of the rightarrow
477
478
p.70, conj 4.18: I don't see how #A_i(Q) = #Phi_{A_i} could possibly hold,
479
given that the former can be infinite, for instance when p=37.
480
481
p.72, Table 4.3: where's 67 ???
482
483
484
485
OK, I'm done (except for section 3.11 on which you wrote
486
"This section has been rewritten").
487
488
489
490
491