CoCalc Shared Fileswww / papers / motive_visibility / email-05-31-02.txtOpen in CoCalc with one click!
Author: William A. Stein
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Comments. 05-31-02
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Most are small stylistic comments; the ones with nontrivial
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mathematical content are set off by *'s.
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* page 1: Change "with which are associated modular forms" to
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"which are associated to modular forms". It sounds less awkard
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and exhibits better parallelism with the second part of the sentence.
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* page 1, last para: change "are congruent modulo some q" to
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"are congruent modulo a maximal ideal q of odd residue characteristic".
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Otherwise it is not true that L(g,s) is forced to vanish to order >= 2.
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Also, say that f and g are forms for Gamma_0. Alternatively, say
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"It turns out, under additional hypothesis, that this forces..."
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* page 1, last para: change "in each case q appears in the" to
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"in each case q divides the numerator".
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* page 2, last para of intro: change "whose existence we cannot prove
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theoretically" to "whose existence we do not prove theoretically."
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You or I might very well eventually be able to prove them, so it's
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not right to say that we "cannot prove" them. All we can see is
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that we do not, in the present paper, prove them.
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* page 2. There is a single occurence of \ldots on this page,
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in the Hodge filtration, and it would be better replaced by a \cdots.
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* page 3, first line: I don't like starting sentences with symbols.
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An easy fix would be to replace '"finite part". D_p is' by
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'"finite part", D_p is', i.e., replace the first period by a comma.
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* page 3, line 5: The subscript on B_dR should be in roman face, but
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is italicized here. Elsewhere in our paper it is correctly typeset
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in roman.
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* page 3, line 10, "Let H_f^1...":
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This should not be a new paragraph.
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* page 4, in the displayed definition of ord_{\lambda}(c_p(j))
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the lambda have turned into q's in two places.
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* page 5, Lemma 4.3: some comments:
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a) The notation "I_p" was only introduced in the proof
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of Lemma 4.2. Since it is used repeatedly, it might
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be better to introduce it before the statement of
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Lemma 4.2.
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b) You use dim a lot, and it's not very ambiguous.
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Nonetheless, it might be nice to say what you view
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the base field that you're taking dimensions as being.
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It's O/q I assume.
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c) * First sentence of proof: I don't see why it suffices
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to prove that equality. *
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* Section 5: I don't remember checking that none of our q divide Phi(N).
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I just checked that now and it's OK. I'm sure you checked it.
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* page 7: Replace "This is analogous to the remark at the end of
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Section 3 of \cite{CM}. It shows that" by "This is analogous to the
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remark at the end of Section 3 of \cite{CM}, which shows that".
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* page 8: Case 2 of the proof, two comments:
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a) I would replace "First we must show that" by
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"First we show that". It is more efficient. Also,
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maybe there is a way to prove this that doesn't first
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begin by showing that H^0 is q-divisible, so maybe
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we don't absolutely have to proceed in that manner.
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b) * You say that equality of dimensions would imply that
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a certain map is injective. Doesn't this assume that
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H^0(I_p, T_q'(k/2)) = 0, and if so, how do we know this? *
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* Did we ever check the hypothesis to Theorem 6.1, that
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"for all p|N, p=/=-w_p (mod q), with p=/=-1(mod q) if p^2|N"?
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I just checked, and amazingly there are no examples (N,q) such
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that a divisor p of N satisfies p = +/-1 (mod q).
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* Page 10, replace "then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not
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dividing" by "then by the sequence of absolute values $|\mbox{\rm
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Tr}(a_p(g))|)$, for~$p$ not dividing"
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* Page 11, replace "standard part of the MAGMA V2.8 (see \cite{magma})."
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by "standard part of MAGMA (see \cite{magma})." That is, delete
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"the" before MAGMA and the version, which is wrong by now anyways.
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* Page 11, replace "X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}" in the
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middle of the page by "X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}".
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* Page 11, in "For the two examples \nf{581k4} and \nf{684k4K}, the square"
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we forgot part of one of the labels; it should be \nf{581k4E}.
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* Page 12: Everyone on this page the spacing in expressions like
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"between $g=$\nf{81k4A}" looks bad. You can put the \nf macro
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in math mode like so: "$g=\nf{81k4A}$", which looks better.
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* Page 12: Lots of modq's which should be "mod q"'s. I.e., there
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is missing space between mod and q. Instead of typesetting this
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with "$\bmod{ \qq}$", instead try "$\bmod{\,\qq}$".
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I still haven't computed the decomposition of q in each Fourier field.
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It's coming up.