From - Fri Jul 7 08:38:06 20001Received: from rancilio.math.Berkeley.EDU (rancilio.Math.Berkeley.EDU [169.229.58.27])2by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id SAA059063for <[email protected]>; Wed, 5 Jul 2000 18:40:59 -0700 (PDT)4From: "Kenneth A. Ribet" <[email protected]>5Received: (from [email protected])6by rancilio.math.Berkeley.EDU (8.9.3/8.9.3) id SAA213247for [email protected]; Wed, 5 Jul 2000 18:40:59 -0700 (PDT)8Date: Wed, 5 Jul 2000 18:40:59 -0700 (PDT)9Message-Id: <[email protected]>10To: [email protected]11Subject: Re: PCMI volume 912Content-Type: text13X-Mozilla-Status: 901114X-Mozilla-Status2: 0000000015X-UIDL: a720bd60920fbcf0b6d7538bf82982e61617Hi William,1819If I write nothing, this usually means that I agree with your20comments/reactions.2122Thanks,23Ken2425----2627> p. 5, middle: independently is misspelled.2829yep, there is a typo.30FIXED3132> p. 7 (-7) Replace question mark with a period.3334yep.35FIXED3637> p. 8 (+1) A precise definition should be given for the phrase38> "mod \ell form" since it is used quite a lot.3940I am not going to write one. Section 1.5.5., "Mod ell modular forms", lists41several references that give the precise definition. Ken, if you feel that42such a definition is needed here, please instruct me as to what to do...4344RIBET: I hope that the index lists "mod ell modular form(s)" and gives45a good page reference for them. I'm pretty sure that it does. Right?4647WAS: It does list "Modular forms | mod ell." I just added a reference to48"Mod $\ell$ | modular form".49FIXED5051> p. 9 (+5) "was", not "were"5253Hmmm. The sentence is "The result, of course, were the conjectures of [10154(serre's duke paper)]." I think we mean "The results, of course, were the55conjectures of [101]."5657RIBET: Best to re-write the sentence. Your solutions sounds good.5859FIXED6061> (+18) Replace colon with period.6263OK. That's sensible.6465> p. 10 (+7) Perhaps explain why this is a concrete consequence,66> saying a little about J_0(Nl^2); also, the end of the line67> sticks out badly into the margin.6869That this is a concrete consequence is explained in Section 3.1 of chapter 3.70Why does Brian want level Nl^2 instead of level Nl?7172RIBET: If you want weights beyond l+1, then you have to twist73by powers of the cyclotomic character, which you view as a character74of conductor l. That twisting raises level from ...l to ...l^2.75Whether Brian is right or not depends on the context -- were76high weights contemplated at that juncture?7778WAS: -- YES -- It says:79A concrete consequence of the conjecture is that all odd80irreducible 2-dimensional~$\rho$ come from abelian varieties81over~$\Q$. Given~$\rho$, one should be able to find a totally82real or CM number field~$E$, an abelian variety~$A$ over~$\Q$83of dimension [...]8485I understand why it is necessary to introduce ell^2. However,86I'm not sure we should explain further, at least at the late point87in the writing of the paper.8889*However* maybe we should mention "Theorem F" (page 4) of Taylor's amazing new90paper "Remarks on a Conjecture of Fontaine and Mazur", which seems to prove the91above concrete consequence under a reasonable local hypothesis.9293OK: I'll fix the jutting-out rho.9495FIXED9697> (+11) Replace colon with period.9899The colon seems appropriate in this context: "... with the following100property:". Ken, what do you think?101102RIBET: I don't know the context, so I give you my proxy.103104105> p. 12106> (-15) Replace E with B.107108OK.109DONE110111> (-1) This table header is badly placed; should be on next page.112113YEP!114115> Aargh. You introduce a notation for the Tate curve without ever116> saying what it is (and the subscript in G_m should not be in boldface117> font). How is the uninitiated reader supposed to interpret the assertion118> that some notation gives rise to an isomorphism? Say that the Tate curve is119> a specific scheme over Z[[q]] such that....120121Yuck. This is a mess.1221) I'll fix the too-bold G_m.123DONE1242) Here's how I propose to reword this. First I'll give the original wording,125then my suggesting for the new wording.126Original wording: "The Tate curve, which we denote by $\Gm/q^\Z$, gives rise127to a $\Gal(\Qpbar/K)$-equivariant isomorphism $E(\Qpbar) \isom128\Qpbar^*/q^\Z$."129My suggested change: "There is a $\Gal(\Qpbar/K)$-equivariant isomorphism130$E(\Qpbar) \isom \Qpbar^*/q^\Z $. The Tate curve, which we suggestively denote131by $\Gm/q^\Z$, is a scheme over $\Z[[q]]$ whose $\Qpbar$ points equal132$\Qpbar^*/q^\Z$."133134RIBET: In that reformulation, the reader has to figure out that the135variable q is mapped to a specific element of Q_p^*, also usually136called q.137138WAS: Good point. How about my suggested changed, but with "over Z[[q]]" omitted.139140> (-1) K*/q^Z is not a p-adic elliptic curve; it is the group of141> rational points of a p-adic (or better: rigid analytic) elliptic curve142> over K.143144Delete the words "p-adic elliptic curve".145146> p. 21 The terminology "supersingular elliptic curve" over Q_l147> (i.e., over a field of char 0) is non-standard, and this should be noted.148149RIBET: A sipersingular curve over a p-adic field is one with good150supersingular reduction. This is somewhat non-standard terminology,151but I think that it's used elsewhere. We can simply say in a sentence152or two what we mean by an ordinary and a supersingular curve over Q_p.153Your solution is ok, too, of course. A curve with additive reduction154is certainly not either supersingular or ordinary, by the way.155156Excerpt from original text: "A \defn{supersingular elliptic curve}~$A$ over157$\Ql$ is an elliptic curve with ...158If~$A$ is not supersingular it is an \defn{ordinary ... In159\cite{serre:propgal}, Serre proved that the representation160$$I_t\ra\Aut(A[\ell])\subset\GL(2,\Fellbar)$$"1611621) A search with grep, and my memory, indicate that we don't use this163terminology anywhere else in our paper.1642) I suggested replacing the two sentences of definitions by: "Let A be an165elliptic curve over $\Ql$ with good reduction such that166$\tilde{A}(\Fellbar)[\ell]=0$." Justification: It seems like the paragraph, as167it is now, makes no sense, because it's not even true when A is ordinary, and168we don't (currently) make it clear that we are only considering supersingular169A.170171RIBET: In this formulation, the "such that...=0" is too wordy.172The reader will know already what good supersingular reduction is.173174WAS: Now it says: "Let~$A$ be an elliptic curve over $\Ql$ with good175supersingular reduction."176177> p. 23 (-7) The concept of mod \ell eigenform has not been defined.178> A reference should be given for the intrinsic defn of theta(f).179180We just said that Katz defines them in [60]. I didn't realize truncating181Emerton's "mod \ell modular forms" section from chapter 1 would cause such a182problem. Argh.183184Ken -- I'm afraid I don't know what Brian means by an intrinsic defn of185theta(f). I don't have any references here with me in Leiden.186187****188RIBET: I presume that he means the definition in terms of the Gauss-189Manin connection. Katz gives this in his paper in LNM volume 601, but190he might wel have given it in his Antwerp paper. (My guess is that he191didn't.)192****193???194195> p. 25 (+2) Note that without loss of generality, k >= 2.196197OK. Our opening sentence is "As a digression, we pause to single out some of198the tools involved in one possible proof of Theorem 2.7."199I suggest inserting the phrase "Note that by twisting we may assume without200loss of generality that k \geq 2."201202> p. 28203>204> (+11) Better to say "see also the appendix" rather than "see also Conrad's205> appendix to this paper", as the latter seems like it says I wrote206> an appendix to Shimura's paper [105].207208OK: But, since Kevin wrote an appendix we should say: "see also Conrad's209appendix".210211> In section 2.3.1.1, replace E_N with E^{\rm{sm}} (the smooth locus of E);212> since E is typically not a group scheme. Also, replace213> E[p] with E^{\rm{sm}}[p].214215OK: This seems like a good idea to me. I forgot about the cusps! I wonder216what notation Gross uses in his paper?217218> (-4) Writing X_1(N)_{/\Q} is clearer.219220OK. This is where we are defining T_p on divisors on the algebraic curve221X_1(N).222223> (-2) "non-cuspidal $\Qbar$-point..."224225OK, as it is more precise.226227> (-1) Aargh. Write E', not \varphi E. It's more natural.228229Hmm. OK. \varphi E(\Qbar) makes good sense, but not \varphi E.230231> p. 29232>233> (+3) The notation Pic^0(X_1(N)) is obscure; it looks like a Picard234> group rather than a scheme. The more standard (and clearer) notation235> is Pic^0_{X_1(N)/Z[1/N]}.236237Hmm. Ken, what do you think. I'm neutral on this.238239RIBET: Can you use words instead of symbols?240241WAS: How about: "This map on divisors242defines an endomorphism $T_p$ of the Jacobian $J_1(N)$ associated243to $X_1(N)$ via Picard functoriality."244245246> (+6) The definition of <d> works perfectly well without smoothness247> conditions. Hence, replace "operator. On non-cuspidal points"248> with "operator, defined functorially by"249> (as this works for relative generalized elliptic curves too).250251OK. Looks good. I'm not 100% this is right though.252253> (+8) The notation J_1(N) should be defined (e.g., is it254> Pic^0...?). It seems to me that the definitions given here255> are inconsistent with those in the appendix; there appears256> to be confusion with respect to issues of Pic vs Alb.257258Hmm.2591) We did define J_1(N) on line 3.2602) They may be inconsistent, but we can't change now, though it would be good261to point out the incosistency.262263RIBET: As you know, I like J_1 to be the Picard variety of X_1.264265> line 2 of section 2.3.1.2 "Recall that A = A_f is..."266267OK. He just means to add "=A_f", which seems like a good idea.268269> pp. 42-43 Pictures for Figures 1 and 2 are missing.270271No worries -- they're postscript and he got only the dvi file...272273> p. 42, section 3.8, paragraph 1. C'mon, anyone who doesn't know274> what a scheme is shouldn't be reading this article. Cut the whole275> paragraph. In the next paragraph, it should be explicitly stated276> how minimal primes and maximal primes of T are to be interpreted277> in the manner indicated.278279Hmm. I disagree with his suggestion to cut the first paragraph. Also, it's280explicitly stated how the diagram corresponds to the various types of primes in281each of the examples.282283> p. 45 Figure 3 is missing.284285No worries.286287> end of 1st paragraph: it is obscure why \calV is288> a finite flat group scheme rather than just a quasi-finite289> flat group scheme. Also, the scheme \calV is not290> 2-dimensional, so the phrase "n-dimensional T/m-vector291> space scheme" should be defined.292293Hmm. I don't have a reference with the exact definition of n-dim'l k-vector294space scheme here, there it's the natural definition. I don't think we use295our assertion that $\calV$ is a "2-dimensional T/m-vector space scheme" later296in the exposition, only the related assertion that "V=\cV_{\Fp}(\Fpbar)". We297could put "2-dimension T/m-vector space scheme" in quotes, omit it, or define298it precisely? Ken, what do you think?299300RIBET: It's defined very well in Raynaud's paper in the Bull of the SMF.301I think that it's just a commutative group scheme V plus a map of302the field into End(V). But I don't have the paper here with me either!!303304WAS: I'll try to look it up tomorrow...305306> p. 63307>308> In the first paragraph, where I say "serious digression from our309> expository goal", maybe add in "; see [Chapter~3]{my book} for details."310> where "my book" = the book I've quasi-written on Ramanujan conjecture311> (to be described as "in preparation").312313OK. I have the exact reference here.314315RIBT: I hope that Brian publishes his book with Springer!!316317> On bottom of page there is a very bad line break.318319OK. Easy to fix.320321--------322Yay! -ken323324325326