CoCalc Shared Fileswww / papers / pcmi / email-jul07.txt
Author: William A. Stein
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5From: "Kenneth A. Ribet" <[email protected]>
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9Date: Wed, 5 Jul 2000 18:40:59 -0700 (PDT)
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11To: [email protected]
12Subject: Re: PCMI volume 9
13Content-Type: text
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17
18Hi William,
19
20If I write nothing, this usually means that I agree with your
22
23Thanks,
24Ken
25
26----
27
28> p. 5, middle:  independently is misspelled.
29
30yep, there is a typo.
31FIXED
32
33> p. 7 (-7) Replace question mark with a period.
34
35yep.
36FIXED
37
38> p. 8 (+1) A precise definition should be given for the phrase
39> "mod \ell form" since it is used quite a lot.
40
41I am not going to write one.   Section 1.5.5., "Mod ell modular forms", lists
42several references that give the precise definition.   Ken, if you feel that
43such a definition is needed here, please instruct me as to what to do...
44
45RIBET: I hope that the index lists "mod ell modular form(s)" and gives
46a good page reference for them.  I'm pretty sure that it does.  Right?
47
48WAS: It does list "Modular forms | mod ell."  I just added a reference to
49"Mod $\ell$ | modular form".
50FIXED
51
52> p. 9 (+5) "was", not "were"
53
54Hmmm.   The sentence is "The result, of course, were the conjectures of [101
55(serre's duke paper)]."  I think we mean "The results, of course, were the
56conjectures of [101]."
57
58RIBET: Best to re-write the sentence.  Your solutions sounds good.
59
60FIXED
61
62> (+18) Replace colon with period.
63
64OK. That's sensible.
65
66> p. 10 (+7) Perhaps explain why this is a concrete consequence,
67> saying a little about J_0(Nl^2); also, the end of the line
68> sticks out badly into the margin.
69
70That this is a concrete consequence is explained in Section 3.1 of chapter 3.
71Why does Brian want level Nl^2 instead of level Nl?
72
73RIBET: If you want weights beyond l+1, then you have to twist
74by powers of the cyclotomic character, which you view as a character
75of conductor l.  That twisting raises level from ...l to ...l^2.
76Whether Brian is right or not depends on the context -- were
77high weights contemplated at that juncture?
78
79WAS: -- YES -- It says:
80   A concrete consequence of the conjecture is that all odd
81   irreducible 2-dimensional~$\rho$ come from abelian varieties
82   over~$\Q$.  Given~$\rho$, one should be able to find a totally
83   real or CM number field~$E$, an abelian variety~$A$ over~$\Q$
84   of dimension [...]
85
86I understand why it is necessary to introduce ell^2.  However,
87I'm not sure we should explain further, at least at the late point
88in the writing of the paper.
89
90*However* maybe we should mention "Theorem F" (page 4) of Taylor's amazing new
91paper "Remarks on a Conjecture of Fontaine and Mazur", which seems to prove the
92above concrete consequence under a reasonable local hypothesis.
93
94OK: I'll fix the jutting-out rho.
95
96FIXED
97
98> (+11) Replace colon with period.
99
100The colon seems appropriate in this context: "... with the following
101property:".   Ken, what do you think?
102
103RIBET: I don't know the context, so I give you my proxy.
104
105
106> p. 12
107> (-15) Replace E with B.
108
109OK.
110DONE
111
112> (-1) This table header is badly placed; should be on next page.
113
114YEP!
115
116> Aargh. You introduce a notation for the Tate curve without ever
117> saying what it is (and the subscript in G_m should not be in boldface
118> font).  How is the uninitiated reader supposed to interpret the assertion
119> that some notation gives rise to an isomorphism?  Say that the Tate curve is
120> a specific scheme over Z[[q]] such that....
121
122Yuck.   This is a mess.
1231) I'll fix the too-bold G_m.
124  DONE
1252) Here's how I propose to reword this.  First I'll give the original wording,
126then my suggesting for the new wording.
127Original wording:  "The Tate curve, which we denote by $\Gm/q^\Z$, gives rise
128to  a $\Gal(\Qpbar/K)$-equivariant isomorphism $E(\Qpbar) \isom 129\Qpbar^*/q^\Z$."
130My suggested change:  "There is a $\Gal(\Qpbar/K)$-equivariant isomorphism
131$E(\Qpbar) \isom \Qpbar^*/q^\Z$.  The Tate curve, which we suggestively denote
132by $\Gm/q^\Z$, is a scheme over $\Z[[q]]$ whose $\Qpbar$ points equal
133$\Qpbar^*/q^\Z$."
134
135RIBET: In that reformulation, the reader has to figure out that the
136variable q is mapped to a specific element of Q_p^*, also usually
137called q.
138
139WAS: Good point.  How about my suggested changed, but with "over Z[[q]]" omitted.
140
141> (-1) K*/q^Z is not a p-adic elliptic curve; it is the group of
142> rational points of a p-adic (or better: rigid analytic) elliptic curve
143> over K.
144
145Delete the words "p-adic elliptic curve".
146
147> p. 21 The terminology "supersingular elliptic curve" over Q_l
148> (i.e., over a field of char 0) is non-standard, and this should be noted.
149
150RIBET: A sipersingular curve over a p-adic field is one with good
151supersingular reduction.  This is somewhat non-standard terminology,
152but I think that it's used elsewhere.  We can simply say in a sentence
153or two what we mean by an ordinary and a supersingular curve over Q_p.
154Your solution is ok, too, of course.  A curve with additive reduction
155is certainly not either supersingular or ordinary, by the way.
156
157Excerpt from original text: "A \defn{supersingular elliptic curve}~$A$ over
158$\Ql$ is an elliptic curve with ...
159If~$A$ is not supersingular it is an \defn{ordinary ... In
160\cite{serre:propgal}, Serre proved that  the representation
161   $$I_t\ra\Aut(A[\ell])\subset\GL(2,\Fellbar)$$"
162
1631) A search with grep, and my memory, indicate that we don't use this
164terminology anywhere else in our paper.
1652) I suggested replacing the two sentences of definitions by: "Let A be an
166elliptic curve over $\Ql$ with good reduction such that
167$\tilde{A}(\Fellbar)[\ell]=0$."  Justification: It seems like the paragraph, as
168it is now, makes no sense, because it's not even true when A is ordinary, and
169we don't (currently) make it clear that we are only considering supersingular
170A.
171
172RIBET: In this formulation, the "such that...=0" is too wordy.
173The reader will know already what good supersingular reduction is.
174
175WAS: Now it says: "Let~$A$ be an elliptic curve over $\Ql$ with good
176supersingular reduction."
177
178> p. 23 (-7) The concept of mod \ell eigenform has not been defined.
179> A reference should be given for the intrinsic defn of theta(f).
180
181We just said that Katz defines them in [60].    I didn't realize truncating
182Emerton's "mod \ell modular forms" section from chapter 1 would cause such a
183problem.  Argh.
184
185Ken -- I'm afraid I don't know what Brian means by an intrinsic defn of
186theta(f).  I don't have any references here with me in Leiden.
187
188****
189RIBET: I presume that he means the definition in terms of the Gauss-
190Manin connection.  Katz gives this in his paper in LNM volume 601, but
191he might wel have given it in his Antwerp paper.  (My guess is that he
192didn't.)
193****
194???
195
196> p. 25 (+2) Note that without loss of generality, k >= 2.
197
198OK.  Our opening sentence is "As a digression, we pause to single out some of
199the tools involved in one possible proof of Theorem 2.7."
200I suggest inserting the phrase "Note that by twisting we may assume without
201loss of generality that k \geq 2."
202
203> p. 28
204>
206> appendix to this paper", as the latter seems like it says I wrote
207> an appendix to Shimura's paper [105].
208
209OK: But, since Kevin wrote an appendix we should say: "see also Conrad's
210appendix".
211
212> In section 2.3.1.1, replace E_N with E^{\rm{sm}} (the smooth locus of E);
213> since E is typically not a group scheme.  Also, replace
214> E[p] with E^{\rm{sm}}[p].
215
216OK: This seems like a good idea to me.   I forgot about the cusps!  I wonder
217what notation Gross uses in his paper?
218
219> (-4) Writing X_1(N)_{/\Q} is clearer.
220
221OK.    This is where we are defining T_p on divisors on the algebraic curve
222X_1(N).
223
224> (-2) "non-cuspidal $\Qbar$-point..."
225
226OK, as it is more precise.
227
228> (-1) Aargh.  Write E', not \varphi E.  It's more natural.
229
230Hmm.  OK.  \varphi E(\Qbar) makes good sense, but not \varphi E.
231
232> p. 29
233>
234> (+3) The notation Pic^0(X_1(N)) is obscure; it looks like a Picard
235> group rather than a scheme.  The more standard (and clearer) notation
236> is Pic^0_{X_1(N)/Z[1/N]}.
237
238Hmm.  Ken, what do you think. I'm neutral on this.
239
240RIBET: Can you use words instead of symbols?
241
242WAS: How about: "This map on divisors
243defines an endomorphism $T_p$ of the Jacobian $J_1(N)$ associated
244to $X_1(N)$ via Picard functoriality."
245
246
247> (+6) The definition of <d> works perfectly well without smoothness
248> conditions.  Hence, replace "operator.  On non-cuspidal points"
249> with "operator, defined functorially by"
250> (as this works for relative generalized elliptic curves too).
251
252OK. Looks good.   I'm not 100% this is right though.
253
254> (+8) The notation J_1(N) should be defined (e.g., is it
255> Pic^0...?).  It seems to me that the definitions given here
256> are inconsistent with those in the appendix; there appears
257> to be confusion with respect to issues of Pic vs Alb.
258
259Hmm.
2601) We did define J_1(N) on line 3.
2612) They may be inconsistent, but we can't change now, though it would be good
262to point out the incosistency.
263
264RIBET: As you know, I like J_1 to be the Picard variety of X_1.
265
266> line 2 of section 2.3.1.2 "Recall that A = A_f is..."
267
268OK.  He just means to add "=A_f", which seems like a good idea.
269
270> pp. 42-43  Pictures for Figures 1 and 2 are missing.
271
272No worries -- they're postscript and he got only the dvi file...
273
274> p. 42, section 3.8, paragraph 1. C'mon, anyone who doesn't know
275> what a scheme is shouldn't be reading this article.  Cut the whole
276> paragraph.  In the next paragraph, it should be explicitly stated
277> how minimal primes and maximal primes of T are to be interpreted
278> in the manner indicated.
279
280Hmm.   I disagree with his suggestion to cut the first paragraph.    Also, it's
281explicitly stated how the diagram corresponds to the various types of primes in
282each of the examples.
283
284> p. 45 Figure 3 is missing.
285
286No worries.
287
288> end of 1st paragraph:  it is obscure why \calV is
289> a finite flat group scheme rather than just a quasi-finite
290> flat group scheme.  Also, the scheme \calV is not
291> 2-dimensional, so the phrase "n-dimensional T/m-vector
292> space scheme" should be defined.
293
294Hmm.  I don't have a reference with the exact definition of n-dim'l k-vector
295space scheme here, there it's the natural definition.    I don't think we use
296our assertion that $\calV$ is a "2-dimensional T/m-vector space scheme" later
297in the exposition, only the related assertion that "V=\cV_{\Fp}(\Fpbar)".   We
298could put "2-dimension T/m-vector space scheme" in quotes, omit it, or define
299it precisely?  Ken, what do you think?
300
301RIBET: It's defined very well in Raynaud's paper in the Bull of the SMF.
302I think that it's just a commutative group scheme V plus a map of
303the field into End(V).  But I don't have the paper here with me either!!
304
305WAS: I'll try to look it up tomorrow...
306
307> p. 63
308>
309> In the first paragraph, where I say "serious digression from our
310> expository goal", maybe add in "; see [Chapter~3]{my book} for details."
311> where "my book" = the book I've quasi-written on Ramanujan conjecture
312> (to be described as "in preparation").
313
314OK.  I have the exact reference here.
315
316RIBT: I hope that Brian publishes his book with Springer!!
317
318> On bottom of page there is a very bad line break.
319
320OK. Easy to fix.
321
322--------
323Yay!  -ken
324
325
326