 CoCalc Shared Fileswww / papers / motive_visibility / email-05-31-02.txt
Author: William A. Stein
1Comments. 05-31-02
2
3Most are small stylistic comments; the ones with nontrivial
4mathematical content are set off by *'s.
5
6* page 1: Change "with which are associated modular forms" to
7  "which are associated to modular forms".  It sounds less awkard
8  and exhibits better parallelism with the second part of the sentence.
9
10* page 1, last para: change "are congruent modulo some q" to
11  "are congruent modulo a maximal ideal q of odd residue characteristic".
12  Otherwise it is not true that L(g,s) is forced to vanish to order >= 2.
13  Also, say that f and g are forms for Gamma_0.  Alternatively, say
14  "It turns out, under additional hypothesis, that this forces..."
15
16* page 1, last para: change "in each case q appears in the" to
17  "in each case q divides the numerator".
18
19* page 2, last para of intro: change "whose existence we cannot prove
20  theoretically" to "whose existence we do not prove theoretically."
21  You or I might very well eventually be able to prove them, so it's
22  not right to say that we "cannot prove" them.  All we can see is
23  that we do not, in the present paper, prove them.
24
25* page 2. There is a single occurence of \ldots on this page,
26  in the Hodge filtration, and it would be better replaced by a \cdots.
27
28* page 3, first line: I don't like starting sentences with symbols.
29  An easy fix would be to replace '"finite part". D_p is' by
30  '"finite part", D_p is', i.e., replace the first period by a comma.
31
32* page 3, line 5: The subscript on B_dR should be in roman face, but
33  is italicized here.  Elsewhere in our paper it is correctly typeset
34  in roman.
35
36* page 3, line 10, "Let H_f^1...":
37  This should not be a new paragraph.
38
39* page 4, in the displayed definition of ord_{\lambda}(c_p(j))
40  the lambda have turned into q's in two places.
41
42* page 5, Lemma 4.3:  some comments:
43          a) The notation "I_p" was only introduced in the proof
44             of Lemma 4.2.  Since it is used repeatedly, it might
45             be better to introduce it before the statement of
46             Lemma 4.2.
47          b) You use dim a lot, and it's not very ambiguous.
48             Nonetheless, it might be nice to say what you view
49             the base field that you're taking dimensions as being.
50             It's O/q I assume.
51          c) * First sentence of proof: I don't see why it suffices
52               to prove that equality. *
53
54* Section 5: I don't remember checking that none of our q divide Phi(N).
55  I just checked that now and it's OK.  I'm sure you checked it.
56
57* page 7: Replace "This is analogous to the remark at the end of
58Section 3 of \cite{CM}. It shows that" by "This is analogous to the
59remark at the end of Section 3 of \cite{CM}, which shows that".
60
61* page 8: Case 2 of the proof, two comments:
62          a) I would replace "First we must show that" by
63             "First we show that".  It is more efficient.  Also,
64             maybe there is a way to prove this that doesn't first
65             begin by showing that H^0 is q-divisible, so maybe
66             we don't absolutely have to proceed in that manner.
67          b) * You say that equality of dimensions would imply that
68             a certain map is injective.  Doesn't this assume that
69             H^0(I_p, T_q'(k/2)) = 0, and if so, how do we know this? *
70
71* Did we ever check the hypothesis to Theorem 6.1, that
72  "for all p|N, p=/=-w_p (mod q), with p=/=-1(mod q) if p^2|N"?
73  I just checked, and amazingly there are no examples (N,q) such
74  that a divisor p of N satisfies p = +/-1 (mod q).
75
76* Page 10, replace "then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not
77  dividing" by "then by the sequence of absolute values $|\mbox{\rm 78 Tr}(a_p(g))|)$, for~$p$ not dividing"
79
80* Page 11, replace "standard part of the MAGMA V2.8 (see \cite{magma})."
81  by "standard part of MAGMA (see \cite{magma})."  That is, delete
82  "the" before MAGMA and the version, which is wrong by now anyways.
83
84* Page 11, replace "X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}" in the
85  middle of the page by "X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}".
86
87* Page 11, in "For the two examples \nf{581k4} and \nf{684k4K}, the square"
88  we forgot part of one of the labels; it should be \nf{581k4E}.
89
90* Page 12: Everyone on this page the spacing in expressions like
91  "between $g=$\nf{81k4A}" looks bad.  You can put the \nf macro
92  in math mode like so: "$g=\nf{81k4A}$", which looks better.
93
94* Page 12: Lots of modq's which should be "mod q"'s.  I.e., there
95  is missing space between mod and q.  Instead of typesetting this
96  with "$\bmod{ \qq}$", instead try "$\bmod{\,\qq}$".
97
98I still haven't computed the decomposition of q in each Fourier field.
99It's coming up.