CoCalc Public Fileswww / papers / thesis / hendrik-2.txt
Author: William A. Stein
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15Subject: May 6
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21Dear William,
22	Welcome back from Australia! I hope you had a good time here.
23	Yes, I had several further looks at your thesis. However, it
24is hard to find the time to type them in. When I am sitting at my
25computer there always seems to be something that needs to happen
26first. At any rate, I am now coming in on a weekend and get something
27done.
28	I have been reading big chunks and found them pleasantly
29written. My remarks are mostly cosmetic. They refer to a version I
30printed a while ago. I hope my references are clear, if not just ask.
31
32Somewhere in sec 1.2 you have v-1'. What is a place minus one?! A
33little later
34in unramified -> is unramified
35-- added a remark "Next suppose that~$J$ has good reduction at~$v$
36and that~$v$ is {\em odd}, in the sense that the
37residue characteristic of~$v$ is odd.  To simplify notation in
38this paragraph, since~$v$ is a non-archimedean place
39of $\Q$, we will also let~$v$ denote the odd prime number
40which is the residue characteristic of~$v$."
41
42
43Section 1.3. I find you explain the contents of the tables badly.
44What is (line 6 of sec 1.3) the message that you are trying to
45convey with the bold face display
46		N  isogeny class
47? I see (in that first column) a number followed by a letter. How
48do the letter and the isogeny class determine each other? By means of
49some standard table? Which?
50	The least the reader can expect is that you explain what the
51symbols in the table header mean. And you want to try and give a
52description of which cases ARE in the table. When you say most ...
532593 ... 2161', are there also cases left out below 2161? You are very
54vague. And, what you write about the fourth column seems to refer to
55the fifth.
57>fixed.
58
59	Sec 1.3.1: why is this a justification'? You seem to say there
60are very few of rank greater than 0. That would seem to be e reason to
61include them, not to omit them.
62
63> I changed it to an *example*
64
65Spacing second line of chapter 2 is very ugly.
66
67> I had a "Merel ~\cite"!!
68
69Def 2.1: is  Z[epsilon]  defined?
70
72
73Just before Th 2.6: anitholomorphic -> anti...
74
75> thanks!
76
77Early on in sec 2.4.3: a sentence with two also''s in quick succession.
78
80
81Prop. 2.13: well-defined (IN prop) or well defined (first line proof)? Be
82consistent. That same 1st sentence of the proof has one that' too many.
83
84> See my email defense of the subtle use of hyphens depending on gramatical
85> context.
86
87Just before Prop. 2.25: He gives ...': who is He'? The previous sentence
88makes no mention of a male person.
89> oops! I was referring to Cremona by his reference.
90
91End sec. 2.6: you say plus or minus quotient'. Earlier it was plus one'
92and minus one'.
93> thanks.
94
95Second paragraph 2.7: the sentence The reader ...' doesn't run, and in
96the next sentence our' = ours'?
97
98> ouch; i butchered that par...
99
100First line proof 2.29: separate 2.6 from \cal S; and next line
101defined -> define
102
103> ok.
104
105Algorithm 3.2, step 1: algebra over =? do you mean arithmetic in?
106
107> I meant "algebra over Z/NZ".  However, I can see how that might
108> sound funny to some people and it is not precise, so I've change
109> to "arithmetic in".
110
111And in step 2: does sigma act on THE SET of Manin symbols?
112
113> yep; that's better.
114
115Def 3.10: necessary to define at THIS stage? We have seen epsilon already
116many times. What is  n  in Lemma 3.11?
117
118> I changed to "Recall that a Dirichlet char. is..."  I also got rid
119> of "continuous" which is silly since (Z/NZ)^* has the discrete topology!
120
121A bit later: why parentheses in  (n')th?
122
123> Because Bjorn complained!  At first I didn't have paranthesis.
124> Now I'll just change n' to "m".
125
126classes of mod p Dirichlet characters are in bijection with ...: the SET
127of such IS in bijection with the set ... .
128
129> got it.
130
131
132In def 3.13 restrict  M  to divisors of  N.
133> Woops!!
134
135Why repeat the def in Th 3.14?
136> Basically because I'm copying the theorem pretty mch verbatim
137> from [13].  What do you think?  It's maybe helpful if someone didn't
138> see defn. 13...
139
140
141Sec 3.6.5, display  tr(a_..) lacks comma after  tr(a_5)
142> got it.
143
144before the display with the signs:
145with - corresponding to -> with $-$ corresponding to
146> done.
147
148End of sec 3.7: is through its action: omit is'.
149> done.
150
151End sec 3.9, beinning 3.10: isogeneous', I'd write isogenous' (since you
152don't pronounce it like homogeneous').
153> got it, and found two more instances using grep.
154
155Just after Def 3.34, separate k=2 from theta_f.
156> done, by inserting "the map"
157
158Lemma 3.40: need a condition to guarantee that  tau_i(L or M) are lattices.
159(Is that condition satisfied in your application?)
160> I see -- tau_i could fail to be injective on V, but still
161> be injective on L.
162> In my application I *do* know that the tau_i(L) are lattices.
163
164
165Early sec 3.12: Essentialy -> Essentially
166> woops!
167
168Proof of Th 3.42, As in Mazur's proof of ': something lacking.
169> woops!
170
171In sec. 3.13 you talk about  f  and say  we do not assume that  g  is an
172eigenform', but there is no  g.
173> changed "g" to "f"
174
175In 3.13.1: respect -> respects
176> got it
177
178Alg 3.47: 1/cN -> 1/(cN)
179> got it
180
181Proof of 3.48: too much space before the footnote-2.
182> fixed with an evil negative space.
183
1843.13.7: Fourier coefficient -> Fourier coefficients
185> got it
186
1873.13.8, beginning: the the -> the
188> got it.
189
190Chapter 4 still needs to be looked at.
191	I did not proofread the present message, if there are
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