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