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From: "H.W. Lenstra"
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Subject: May 6
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Dear William,
Welcome back from Australia! I hope you had a good time here.
Yes, I had several further looks at your thesis. However, it
is hard to find the time to type them in. When I am sitting at my
computer there always seems to be something that needs to happen
first. At any rate, I am now coming in on a weekend and get something
done.
I have been reading big chunks and found them pleasantly
written. My remarks are mostly cosmetic. They refer to a version I
printed a while ago. I hope my references are clear, if not just ask.
Somewhere in sec 1.2 you have `v-1'. What is a place minus one?! A
little later
in unramified -> is unramified
-- added a remark "Next suppose that~$J$ has good reduction at~$v$
and that~$v$ is {\em odd}, in the sense that the
residue characteristic of~$v$ is odd. To simplify notation in
this paragraph, since~$v$ is a non-archimedean place
of $\Q$, we will also let~$v$ denote the odd prime number
which is the residue characteristic of~$v$."
Section 1.3. I find you explain the contents of the tables badly.
What is (line 6 of sec 1.3) the message that you are trying to
convey with the bold face display
N isogeny class
? I see (in that first column) a number followed by a letter. How
do the letter and the isogeny class determine each other? By means of
some standard table? Which?
The least the reader can expect is that you explain what the
symbols in the table header mean. And you want to try and give a
description of which cases ARE in the table. When you say `most ...
2593 ... 2161', are there also cases left out below 2161? You are very
vague. And, what you write about the fourth column seems to refer to
the fifth.
Please reconsider everything you say about the tables.
>fixed.
Sec 1.3.1: why is this a `justification'? You seem to say there
are very few of rank greater than 0. That would seem to be e reason to
include them, not to omit them.
> I changed it to an *example*
Spacing second line of chapter 2 is very ugly.
> I had a "Merel ~\cite"!!
Def 2.1: is Z[epsilon] defined?
> Bjorne already made me fix this.
Just before Th 2.6: anitholomorphic -> anti...
> thanks!
Early on in sec 2.4.3: a sentence with two `also''s in quick succession.
> found already...
Prop. 2.13: well-defined (IN prop) or well defined (first line proof)? Be
consistent. That same 1st sentence of the proof has one `that' too many.
> See my email defense of the subtle use of hyphens depending on gramatical
> context.
Just before Prop. 2.25: `He gives ...': who is `He'? The previous sentence
makes no mention of a male person.
> oops! I was referring to Cremona by his reference.
End sec. 2.6: you say `plus or minus quotient'. Earlier it was `plus one'
and `minus one'.
> thanks.
Second paragraph 2.7: the sentence `The reader ...' doesn't run, and in
the next sentence `our' = `ours'?
> ouch; i butchered that par...
First line proof 2.29: separate 2.6 from \cal S; and next line
defined -> define
> ok.
Algorithm 3.2, step 1: algebra over =? do you mean arithmetic in?
> I meant "algebra over Z/NZ". However, I can see how that might
> sound funny to some people and it is not precise, so I've change
> to "arithmetic in".
And in step 2: does sigma act on THE SET of Manin symbols?
> yep; that's better.
Def 3.10: necessary to define at THIS stage? We have seen epsilon already
many times. What is n in Lemma 3.11?
> I changed to "Recall that a Dirichlet char. is..." I also got rid
> of "continuous" which is silly since (Z/NZ)^* has the discrete topology!
A bit later: why parentheses in (n')th?
> Because Bjorn complained! At first I didn't have paranthesis.
> Now I'll just change n' to "m".
classes of mod p Dirichlet characters are in bijection with ...: the SET
of such IS in bijection with the set ... .
> got it.
In def 3.13 restrict M to divisors of N.
> Woops!!
Why repeat the def in Th 3.14?
> Basically because I'm copying the theorem pretty mch verbatim
> from [13]. What do you think? It's maybe helpful if someone didn't
> see defn. 13...
Sec 3.6.5, display tr(a_..) lacks comma after tr(a_5)
> got it.
before the display with the signs:
with - corresponding to -> with $-$ corresponding to
> done.
End of sec 3.7: is through its action: omit `is'.
> done.
End sec 3.9, beinning 3.10: `isogeneous', I'd write `isogenous' (since you
don't pronounce it like `homogeneous').
> got it, and found two more instances using grep.
Just after Def 3.34, separate k=2 from theta_f.
> done, by inserting "the map"
Lemma 3.40: need a condition to guarantee that tau_i(L or M) are lattices.
(Is that condition satisfied in your application?)
> I see -- tau_i could fail to be injective on V, but still
> be injective on L.
> In my application I *do* know that the tau_i(L) are lattices.
Early sec 3.12: Essentialy -> Essentially
> woops!
Proof of Th 3.42, `As in Mazur's proof of ': something lacking.
> woops!
In sec. 3.13 you talk about f and say `we do not assume that g is an
eigenform', but there is no g.
> changed "g" to "f"
In 3.13.1: respect -> respects
> got it
Alg 3.47: 1/cN -> 1/(cN)
> got it
Proof of 3.48: too much space before the footnote-2.
> fixed with an evil negative space.
3.13.7: Fourier coefficient -> Fourier coefficients
> got it
3.13.8, beginning: the the -> the
> got it.
Chapter 4 still needs to be looked at.
I did not proofread the present message, if there are
obscurities just ask.