From - Sat May 6 07:08:16 20001Received: from zada.math.leidenuniv.nl (IDENT:[email protected] [132.229.231.3])2by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id GAA255513for <[email protected]>; Sat, 6 May 2000 06:51:41 -0700 (PDT)4Received: from math.LeidenUniv.nl (IDENT:r[email protected] [132.229.232.66])5by zada.math.leidenuniv.nl (8.9.3/8.9.3) with ESMTP id PAA044796for <[email protected]>; Sat, 6 May 2000 15:51:41 +02007Received: (from [email protected])8by math.LeidenUniv.nl (8.9.3/8.9.3) id PAA022479for [email protected]; Sat, 6 May 2000 15:51:41 +020010Date: Sat, 6 May 2000 15:51:41 +020011From: "H.W. Lenstra" <[email protected]>12Message-Id: <[email protected]>13To: [email protected]14Subject: May 615Content-Type: text16X-Mozilla-Status: 800117X-Mozilla-Status2: 0000000018X-UIDL: 0163696c8ce9606eb0156c1f4f12a4d71920Dear William,21Welcome back from Australia! I hope you had a good time here.22Yes, I had several further looks at your thesis. However, it23is hard to find the time to type them in. When I am sitting at my24computer there always seems to be something that needs to happen25first. At any rate, I am now coming in on a weekend and get something26done.27I have been reading big chunks and found them pleasantly28written. My remarks are mostly cosmetic. They refer to a version I29printed a while ago. I hope my references are clear, if not just ask.3031Somewhere in sec 1.2 you have `v-1'. What is a place minus one?! A32little later33in unramified -> is unramified34-- added a remark "Next suppose that~$J$ has good reduction at~$v$35and that~$v$ is {\em odd}, in the sense that the36residue characteristic of~$v$ is odd. To simplify notation in37this paragraph, since~$v$ is a non-archimedean place38of $\Q$, we will also let~$v$ denote the odd prime number39which is the residue characteristic of~$v$."404142Section 1.3. I find you explain the contents of the tables badly.43What is (line 6 of sec 1.3) the message that you are trying to44convey with the bold face display45N isogeny class46? I see (in that first column) a number followed by a letter. How47do the letter and the isogeny class determine each other? By means of48some standard table? Which?49The least the reader can expect is that you explain what the50symbols in the table header mean. And you want to try and give a51description of which cases ARE in the table. When you say `most ...522593 ... 2161', are there also cases left out below 2161? You are very53vague. And, what you write about the fourth column seems to refer to54the fifth.55Please reconsider everything you say about the tables.56>fixed.5758Sec 1.3.1: why is this a `justification'? You seem to say there59are very few of rank greater than 0. That would seem to be e reason to60include them, not to omit them.6162> I changed it to an *example*6364Spacing second line of chapter 2 is very ugly.6566> I had a "Merel ~\cite"!!6768Def 2.1: is Z[epsilon] defined?6970> Bjorne already made me fix this.7172Just before Th 2.6: anitholomorphic -> anti...7374> thanks!7576Early on in sec 2.4.3: a sentence with two `also''s in quick succession.7778> found already...7980Prop. 2.13: well-defined (IN prop) or well defined (first line proof)? Be81consistent. That same 1st sentence of the proof has one `that' too many.8283> See my email defense of the subtle use of hyphens depending on gramatical84> context.8586Just before Prop. 2.25: `He gives ...': who is `He'? The previous sentence87makes no mention of a male person.88> oops! I was referring to Cremona by his reference.8990End sec. 2.6: you say `plus or minus quotient'. Earlier it was `plus one'91and `minus one'.92> thanks.9394Second paragraph 2.7: the sentence `The reader ...' doesn't run, and in95the next sentence `our' = `ours'?9697> ouch; i butchered that par...9899First line proof 2.29: separate 2.6 from \cal S; and next line100defined -> define101102> ok.103104Algorithm 3.2, step 1: algebra over =? do you mean arithmetic in?105106> I meant "algebra over Z/NZ". However, I can see how that might107> sound funny to some people and it is not precise, so I've change108> to "arithmetic in".109110And in step 2: does sigma act on THE SET of Manin symbols?111112> yep; that's better.113114Def 3.10: necessary to define at THIS stage? We have seen epsilon already115many times. What is n in Lemma 3.11?116117> I changed to "Recall that a Dirichlet char. is..." I also got rid118> of "continuous" which is silly since (Z/NZ)^* has the discrete topology!119120A bit later: why parentheses in (n')th?121122> Because Bjorn complained! At first I didn't have paranthesis.123> Now I'll just change n' to "m".124125classes of mod p Dirichlet characters are in bijection with ...: the SET126of such IS in bijection with the set ... .127128> got it.129130131In def 3.13 restrict M to divisors of N.132> Woops!!133134Why repeat the def in Th 3.14?135> Basically because I'm copying the theorem pretty mch verbatim136> from [13]. What do you think? It's maybe helpful if someone didn't137> see defn. 13...138139140Sec 3.6.5, display tr(a_..) lacks comma after tr(a_5)141> got it.142143before the display with the signs:144with - corresponding to -> with $-$ corresponding to145> done.146147End of sec 3.7: is through its action: omit `is'.148> done.149150End sec 3.9, beinning 3.10: `isogeneous', I'd write `isogenous' (since you151don't pronounce it like `homogeneous').152> got it, and found two more instances using grep.153154Just after Def 3.34, separate k=2 from theta_f.155> done, by inserting "the map"156157Lemma 3.40: need a condition to guarantee that tau_i(L or M) are lattices.158(Is that condition satisfied in your application?)159> I see -- tau_i could fail to be injective on V, but still160> be injective on L.161> In my application I *do* know that the tau_i(L) are lattices.162163164Early sec 3.12: Essentialy -> Essentially165> woops!166167Proof of Th 3.42, `As in Mazur's proof of ': something lacking.168> woops!169170In sec. 3.13 you talk about f and say `we do not assume that g is an171eigenform', but there is no g.172> changed "g" to "f"173174In 3.13.1: respect -> respects175> got it176177Alg 3.47: 1/cN -> 1/(cN)178> got it179180Proof of 3.48: too much space before the footnote-2.181> fixed with an evil negative space.1821833.13.7: Fourier coefficient -> Fourier coefficients184> got it1851863.13.8, beginning: the the -> the187> got it.188189Chapter 4 still needs to be looked at.190I did not proofread the present message, if there are191obscurities just ask.192193194