From - Fri Jul 7 08:38:06 2000 Received: from rancilio.math.Berkeley.EDU (rancilio.Math.Berkeley.EDU [169.229.58.27]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id SAA05906 for ; Wed, 5 Jul 2000 18:40:59 -0700 (PDT) From: "Kenneth A. Ribet" Received: (from ribet@localhost) by rancilio.math.Berkeley.EDU (8.9.3/8.9.3) id SAA21324 for was@math.berkeley.edu; Wed, 5 Jul 2000 18:40:59 -0700 (PDT) Date: Wed, 5 Jul 2000 18:40:59 -0700 (PDT) Message-Id: <200007060140.SAA21324@rancilio.math.Berkeley.EDU> To: was@math.berkeley.edu Subject: Re: PCMI volume 9 Content-Type: text X-Mozilla-Status: 9011 X-Mozilla-Status2: 00000000 X-UIDL: a720bd60920fbcf0b6d7538bf82982e6 Hi William, If I write nothing, this usually means that I agree with your comments/reactions. Thanks, Ken ---- > p. 5, middle: independently is misspelled. yep, there is a typo. FIXED > p. 7 (-7) Replace question mark with a period. yep. FIXED > p. 8 (+1) A precise definition should be given for the phrase > "mod \ell form" since it is used quite a lot. I am not going to write one. Section 1.5.5., "Mod ell modular forms", lists several references that give the precise definition. Ken, if you feel that such a definition is needed here, please instruct me as to what to do... RIBET: I hope that the index lists "mod ell modular form(s)" and gives a good page reference for them. I'm pretty sure that it does. Right? WAS: It does list "Modular forms | mod ell." I just added a reference to "Mod $\ell$ | modular form". FIXED > p. 9 (+5) "was", not "were" Hmmm. The sentence is "The result, of course, were the conjectures of [101 (serre's duke paper)]." I think we mean "The results, of course, were the conjectures of [101]." RIBET: Best to re-write the sentence. Your solutions sounds good. FIXED > (+18) Replace colon with period. OK. That's sensible. > p. 10 (+7) Perhaps explain why this is a concrete consequence, > saying a little about J_0(Nl^2); also, the end of the line > sticks out badly into the margin. That this is a concrete consequence is explained in Section 3.1 of chapter 3. Why does Brian want level Nl^2 instead of level Nl? RIBET: If you want weights beyond l+1, then you have to twist by powers of the cyclotomic character, which you view as a character of conductor l. That twisting raises level from ...l to ...l^2. Whether Brian is right or not depends on the context -- were high weights contemplated at that juncture? WAS: -- YES -- It says: A concrete consequence of the conjecture is that all odd irreducible 2-dimensional~$\rho$ come from abelian varieties over~$\Q$. Given~$\rho$, one should be able to find a totally real or CM number field~$E$, an abelian variety~$A$ over~$\Q$ of dimension [...] I understand why it is necessary to introduce ell^2. However, I'm not sure we should explain further, at least at the late point in the writing of the paper. *However* maybe we should mention "Theorem F" (page 4) of Taylor's amazing new paper "Remarks on a Conjecture of Fontaine and Mazur", which seems to prove the above concrete consequence under a reasonable local hypothesis. OK: I'll fix the jutting-out rho. FIXED > (+11) Replace colon with period. The colon seems appropriate in this context: "... with the following property:". Ken, what do you think? RIBET: I don't know the context, so I give you my proxy. > p. 12 > (-15) Replace E with B. OK. DONE > (-1) This table header is badly placed; should be on next page. YEP! > Aargh. You introduce a notation for the Tate curve without ever > saying what it is (and the subscript in G_m should not be in boldface > font). How is the uninitiated reader supposed to interpret the assertion > that some notation gives rise to an isomorphism? Say that the Tate curve is > a specific scheme over Z[[q]] such that.... Yuck. This is a mess. 1) I'll fix the too-bold G_m. DONE 2) Here's how I propose to reword this. First I'll give the original wording, then my suggesting for the new wording. Original wording: "The Tate curve, which we denote by $\Gm/q^\Z$, gives rise to a $\Gal(\Qpbar/K)$-equivariant isomorphism $E(\Qpbar) \isom \Qpbar^*/q^\Z$." My suggested change: "There is a $\Gal(\Qpbar/K)$-equivariant isomorphism $E(\Qpbar) \isom \Qpbar^*/q^\Z $. The Tate curve, which we suggestively denote by $\Gm/q^\Z$, is a scheme over $\Z[[q]]$ whose $\Qpbar$ points equal $\Qpbar^*/q^\Z$." RIBET: In that reformulation, the reader has to figure out that the variable q is mapped to a specific element of Q_p^*, also usually called q. WAS: Good point. How about my suggested changed, but with "over Z[[q]]" omitted. > (-1) K*/q^Z is not a p-adic elliptic curve; it is the group of > rational points of a p-adic (or better: rigid analytic) elliptic curve > over K. Delete the words "p-adic elliptic curve". > p. 21 The terminology "supersingular elliptic curve" over Q_l > (i.e., over a field of char 0) is non-standard, and this should be noted. RIBET: A sipersingular curve over a p-adic field is one with good supersingular reduction. This is somewhat non-standard terminology, but I think that it's used elsewhere. We can simply say in a sentence or two what we mean by an ordinary and a supersingular curve over Q_p. Your solution is ok, too, of course. A curve with additive reduction is certainly not either supersingular or ordinary, by the way. Excerpt from original text: "A \defn{supersingular elliptic curve}~$A$ over $\Ql$ is an elliptic curve with ... If~$A$ is not supersingular it is an \defn{ordinary ... In \cite{serre:propgal}, Serre proved that the representation $$I_t\ra\Aut(A[\ell])\subset\GL(2,\Fellbar)$$" 1) A search with grep, and my memory, indicate that we don't use this terminology anywhere else in our paper. 2) I suggested replacing the two sentences of definitions by: "Let A be an elliptic curve over $\Ql$ with good reduction such that $\tilde{A}(\Fellbar)[\ell]=0$." Justification: It seems like the paragraph, as it is now, makes no sense, because it's not even true when A is ordinary, and we don't (currently) make it clear that we are only considering supersingular A. RIBET: In this formulation, the "such that...=0" is too wordy. The reader will know already what good supersingular reduction is. WAS: Now it says: "Let~$A$ be an elliptic curve over $\Ql$ with good supersingular reduction." > p. 23 (-7) The concept of mod \ell eigenform has not been defined. > A reference should be given for the intrinsic defn of theta(f). We just said that Katz defines them in [60]. I didn't realize truncating Emerton's "mod \ell modular forms" section from chapter 1 would cause such a problem. Argh. Ken -- I'm afraid I don't know what Brian means by an intrinsic defn of theta(f). I don't have any references here with me in Leiden. **** RIBET: I presume that he means the definition in terms of the Gauss- Manin connection. Katz gives this in his paper in LNM volume 601, but he might wel have given it in his Antwerp paper. (My guess is that he didn't.) **** ??? > p. 25 (+2) Note that without loss of generality, k >= 2. OK. Our opening sentence is "As a digression, we pause to single out some of the tools involved in one possible proof of Theorem 2.7." I suggest inserting the phrase "Note that by twisting we may assume without loss of generality that k \geq 2." > p. 28 > > (+11) Better to say "see also the appendix" rather than "see also Conrad's > appendix to this paper", as the latter seems like it says I wrote > an appendix to Shimura's paper [105]. OK: But, since Kevin wrote an appendix we should say: "see also Conrad's appendix". > In section 2.3.1.1, replace E_N with E^{\rm{sm}} (the smooth locus of E); > since E is typically not a group scheme. Also, replace > E[p] with E^{\rm{sm}}[p]. OK: This seems like a good idea to me. I forgot about the cusps! I wonder what notation Gross uses in his paper? > (-4) Writing X_1(N)_{/\Q} is clearer. OK. This is where we are defining T_p on divisors on the algebraic curve X_1(N). > (-2) "non-cuspidal $\Qbar$-point..." OK, as it is more precise. > (-1) Aargh. Write E', not \varphi E. It's more natural. Hmm. OK. \varphi E(\Qbar) makes good sense, but not \varphi E. > p. 29 > > (+3) The notation Pic^0(X_1(N)) is obscure; it looks like a Picard > group rather than a scheme. The more standard (and clearer) notation > is Pic^0_{X_1(N)/Z[1/N]}. Hmm. Ken, what do you think. I'm neutral on this. RIBET: Can you use words instead of symbols? WAS: How about: "This map on divisors defines an endomorphism $T_p$ of the Jacobian $J_1(N)$ associated to $X_1(N)$ via Picard functoriality." > (+6) The definition of works perfectly well without smoothness > conditions. Hence, replace "operator. On non-cuspidal points" > with "operator, defined functorially by" > (as this works for relative generalized elliptic curves too). OK. Looks good. I'm not 100% this is right though. > (+8) The notation J_1(N) should be defined (e.g., is it > Pic^0...?). It seems to me that the definitions given here > are inconsistent with those in the appendix; there appears > to be confusion with respect to issues of Pic vs Alb. Hmm. 1) We did define J_1(N) on line 3. 2) They may be inconsistent, but we can't change now, though it would be good to point out the incosistency. RIBET: As you know, I like J_1 to be the Picard variety of X_1. > line 2 of section 2.3.1.2 "Recall that A = A_f is..." OK. He just means to add "=A_f", which seems like a good idea. > pp. 42-43 Pictures for Figures 1 and 2 are missing. No worries -- they're postscript and he got only the dvi file... > p. 42, section 3.8, paragraph 1. C'mon, anyone who doesn't know > what a scheme is shouldn't be reading this article. Cut the whole > paragraph. In the next paragraph, it should be explicitly stated > how minimal primes and maximal primes of T are to be interpreted > in the manner indicated. Hmm. I disagree with his suggestion to cut the first paragraph. Also, it's explicitly stated how the diagram corresponds to the various types of primes in each of the examples. > p. 45 Figure 3 is missing. No worries. > end of 1st paragraph: it is obscure why \calV is > a finite flat group scheme rather than just a quasi-finite > flat group scheme. Also, the scheme \calV is not > 2-dimensional, so the phrase "n-dimensional T/m-vector > space scheme" should be defined. Hmm. I don't have a reference with the exact definition of n-dim'l k-vector space scheme here, there it's the natural definition. I don't think we use our assertion that $\calV$ is a "2-dimensional T/m-vector space scheme" later in the exposition, only the related assertion that "V=\cV_{\Fp}(\Fpbar)". We could put "2-dimension T/m-vector space scheme" in quotes, omit it, or define it precisely? Ken, what do you think? RIBET: It's defined very well in Raynaud's paper in the Bull of the SMF. I think that it's just a commutative group scheme V plus a map of the field into End(V). But I don't have the paper here with me either!! WAS: I'll try to look it up tomorrow... > p. 63 > > In the first paragraph, where I say "serious digression from our > expository goal", maybe add in "; see [Chapter~3]{my book} for details." > where "my book" = the book I've quasi-written on Ramanujan conjecture > (to be described as "in preparation"). OK. I have the exact reference here. RIBT: I hope that Brian publishes his book with Springer!! > On bottom of page there is a very bad line break. OK. Easy to fix. -------- Yay! -ken