CoCalc Public Fileswww / papers / thesis / hendrik-2.txtOpen with one click!
Author: William A. Stein
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Date: Sat, 6 May 2000 15:51:41 +0200
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From: "H.W. Lenstra" <[email protected]>
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To: [email protected]
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Subject: May 6
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Dear William,
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Welcome back from Australia! I hope you had a good time here.
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Yes, I had several further looks at your thesis. However, it
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is hard to find the time to type them in. When I am sitting at my
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computer there always seems to be something that needs to happen
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first. At any rate, I am now coming in on a weekend and get something
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done.
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I have been reading big chunks and found them pleasantly
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written. My remarks are mostly cosmetic. They refer to a version I
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printed a while ago. I hope my references are clear, if not just ask.
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Somewhere in sec 1.2 you have `v-1'. What is a place minus one?! A
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little later
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in unramified -> is unramified
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-- added a remark "Next suppose that~$J$ has good reduction at~$v$
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and that~$v$ is {\em odd}, in the sense that the
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residue characteristic of~$v$ is odd. To simplify notation in
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this paragraph, since~$v$ is a non-archimedean place
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of $\Q$, we will also let~$v$ denote the odd prime number
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which is the residue characteristic of~$v$."
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Section 1.3. I find you explain the contents of the tables badly.
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What is (line 6 of sec 1.3) the message that you are trying to
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convey with the bold face display
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N isogeny class
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? I see (in that first column) a number followed by a letter. How
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do the letter and the isogeny class determine each other? By means of
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some standard table? Which?
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The least the reader can expect is that you explain what the
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symbols in the table header mean. And you want to try and give a
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description of which cases ARE in the table. When you say `most ...
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2593 ... 2161', are there also cases left out below 2161? You are very
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vague. And, what you write about the fourth column seems to refer to
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the fifth.
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Please reconsider everything you say about the tables.
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>fixed.
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Sec 1.3.1: why is this a `justification'? You seem to say there
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are very few of rank greater than 0. That would seem to be e reason to
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include them, not to omit them.
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> I changed it to an *example*
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Spacing second line of chapter 2 is very ugly.
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> I had a "Merel ~\cite"!!
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Def 2.1: is Z[epsilon] defined?
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> Bjorne already made me fix this.
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Just before Th 2.6: anitholomorphic -> anti...
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> thanks!
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Early on in sec 2.4.3: a sentence with two `also''s in quick succession.
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> found already...
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Prop. 2.13: well-defined (IN prop) or well defined (first line proof)? Be
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consistent. That same 1st sentence of the proof has one `that' too many.
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> See my email defense of the subtle use of hyphens depending on gramatical
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> context.
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Just before Prop. 2.25: `He gives ...': who is `He'? The previous sentence
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makes no mention of a male person.
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> oops! I was referring to Cremona by his reference.
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End sec. 2.6: you say `plus or minus quotient'. Earlier it was `plus one'
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and `minus one'.
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> thanks.
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Second paragraph 2.7: the sentence `The reader ...' doesn't run, and in
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the next sentence `our' = `ours'?
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> ouch; i butchered that par...
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First line proof 2.29: separate 2.6 from \cal S; and next line
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defined -> define
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> ok.
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Algorithm 3.2, step 1: algebra over =? do you mean arithmetic in?
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> I meant "algebra over Z/NZ". However, I can see how that might
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> sound funny to some people and it is not precise, so I've change
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> to "arithmetic in".
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And in step 2: does sigma act on THE SET of Manin symbols?
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> yep; that's better.
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Def 3.10: necessary to define at THIS stage? We have seen epsilon already
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many times. What is n in Lemma 3.11?
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> I changed to "Recall that a Dirichlet char. is..." I also got rid
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> of "continuous" which is silly since (Z/NZ)^* has the discrete topology!
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A bit later: why parentheses in (n')th?
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> Because Bjorn complained! At first I didn't have paranthesis.
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> Now I'll just change n' to "m".
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classes of mod p Dirichlet characters are in bijection with ...: the SET
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of such IS in bijection with the set ... .
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> got it.
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In def 3.13 restrict M to divisors of N.
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> Woops!!
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Why repeat the def in Th 3.14?
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> Basically because I'm copying the theorem pretty mch verbatim
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> from [13]. What do you think? It's maybe helpful if someone didn't
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> see defn. 13...
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Sec 3.6.5, display tr(a_..) lacks comma after tr(a_5)
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> got it.
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before the display with the signs:
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with - corresponding to -> with $-$ corresponding to
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> done.
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End of sec 3.7: is through its action: omit `is'.
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> done.
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End sec 3.9, beinning 3.10: `isogeneous', I'd write `isogenous' (since you
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don't pronounce it like `homogeneous').
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> got it, and found two more instances using grep.
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Just after Def 3.34, separate k=2 from theta_f.
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> done, by inserting "the map"
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Lemma 3.40: need a condition to guarantee that tau_i(L or M) are lattices.
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(Is that condition satisfied in your application?)
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> I see -- tau_i could fail to be injective on V, but still
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> be injective on L.
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> In my application I *do* know that the tau_i(L) are lattices.
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Early sec 3.12: Essentialy -> Essentially
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> woops!
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Proof of Th 3.42, `As in Mazur's proof of ': something lacking.
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> woops!
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In sec. 3.13 you talk about f and say `we do not assume that g is an
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eigenform', but there is no g.
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> changed "g" to "f"
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In 3.13.1: respect -> respects
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> got it
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Alg 3.47: 1/cN -> 1/(cN)
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> got it
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Proof of 3.48: too much space before the footnote-2.
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> fixed with an evil negative space.
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3.13.7: Fourier coefficient -> Fourier coefficients
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> got it
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3.13.8, beginning: the the -> the
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> got it.
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Chapter 4 still needs to be looked at.
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I did not proofread the present message, if there are
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obscurities just ask.
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