Page 3, line -8. There is an undefined reference here, probably because the label for Chapter 5 got removed.
Make sure that all the labels are correctly put in, by having a look at the log file after you've run the file
of the paper through tex, so that there are no ??'s anywhere in the paper.
Page 47,
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line -9. "The local factors at ramified places are described in [Jac]."
It would be good if you could give the more precise reference (theorem or equation number, or such).
Even if it is easy for an expert to make out where the formula for the local factors is
to be found in [Jac], the less experienced reader might be turned off by the prospect of
having to search through a rather big book to find the formula.
Also, on the same line, there is a typo ("idenitified")
The sentence starting with "Indeed, up to a shift,..." while perfectly correct, is a bit heavy.
I propose something like ".. associated to pi_f and pi_chi, where pi_f and pi_chi denote the
automorphic representations attached to f and chi respectively."
Page 48,
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line 4: Same remark as before concerning the reference to [Jac].
line 4 of Case 1: The expression "more precisely" is not really appropriate here: in what way is
"a product of the Shimura periods u+-(f) attached to f" more precise than "the Petersson inner
product "? If anything I would say the latter is *more* precise than the former.
Of course you probably have something more sophisticated in mind, but the sentence reads funny to a non-initiate.
Page 49,
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line 3 (after Figure 1): there should be no comma after "that"
Page 50.
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line 4. I would replace "c| f_chi" by "c divides f_chi."
In general, one should avoid using a mathematical symbol as a verb in a sentence (one of Serre's Golden
Ruled of Good Mathematical Writing Etiquette, which I picked up as a graduate student working on the notes
of his course!)
line 7: comma after "principal"
6th diplayed equation: Here you do not mention anymore the (2') region (which you introduced earlier, and I
guess indeed it is warranted.) Perhaps you should make sure that the notations and statements here are consistent,
i.e., that in Sigma_cc(N) we are ignoring the characters in the (2') region.
Anyway, I guess this is an important distinction because having the three regions makes it clear that there are *three* p-adic L-functions and one might ask what the p-adic L-function interpolating the special values in the
(2) region look like when evaluated at a central critical character in the (2') region --
a question that we do not address in our paper, and perhaps it is worth pointing this out..?
7th displayed equation: right before this equation, "Recall the Shimura-Maass operator of equation (32)"
should instead refer to (31).
Last line, in proof of Lemma 5.4.
The proof is a bit terse, perhaps we should replace the last sentence by
"The lemma then follows from equation (136) with f replaced by delta_k^jf and k by k+2j,
(137), and (142)."
Page 51,
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Note that the formula for C(f,chi,c) still contains an undetermined power of 2 in it!
Also, there is an eta' appearing in that formula, which I think has not been defined yet.
In the section on Differential operators, it seems funny to start the section in this way when the
Shimura-Maass operator has been invoked just one page before. I suggest a line like
"We recall some general facts about the Shimura-Maass operators of (142) introduced in
(31) "
where this would require adding a label to the displayed equation of the previous page (or, leave that out if you think it is too much..."
The second sentence after that, "For the moment we will use the symbol f for an arbitrary modular form"
is a bit vague, since what an arbitrary modular form is depends a lot on the mathematician
who is writing (would you want f to be weakly holomorphic modular form, a nearly holomorphic
modular form, a meromorphic modular form, a rigid analytic modular form, a harmonic modular form,
a Maass form, a weak harmonic Maass form...? ..!)
I would just move that sentence to right after the next displayed equation, and write
"For the moment we will use the symbol f for an arbitrary modular form in C^infty_k(Gamma) or
C^infty_k-tilde(Gamma). "
Equation (144). You should mention that this is the same equation as equation (31) of page (10), except
that the space of modular forms is a bit different, and that r is replaced by k.
Displayed equation right after (144): I have the impression that the symbol \Lambda is a bit
overused, since we use it to denote the completed L-function, and also various lattices
that crop up in the definition of the complex Abel Jacobi map. If it is possible to replace
Lambda by something a bit more distinctive, I think that would be nice (but then, make sure, by making a search for
\Lambda in the entire file, that the notation is being changed everywhere, and to something that is also not already
taken up : there is, unfortunately, already a lot of notation in this paper!)
Right after equation (145), "following the discussion in [Bump], it would be good to at least
add a chapter or section number...
Statement of Lemma 5.7: Suppose *that* f ... is holomorphic. Then for j>=0, *the form* Rj f-tilde in
C-tilde){k+2j}(Gamma) is an eigenfunction..."
THe main point here is that (another of the Serre Golden Rules) one should avoid starting a sentence or
the clause after a comma with a mathematical symbol, because this is usually typographically ugly at best,
and can lead to confusion at worst.
Page 52
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Line 1, you forgot an f-tilde in the equation in that line.
In definition 5.8., you use a boldface H to denote the Poincare upper half plane. Make a search
throughout the file to make sure that the notations are consistent. (I would prefer to use \cH,
meaning {\cal H}, everywhere to denote the Poincare UHP...)
Right after Defintion 5.8., typo: "begin" shoud be "being"
Proof of Lemma 5.9: This proof is terse to the point of being hard to read.
It is fine to invoke, in a single sentence, a lemma, an equation, and an outside citation, to
justify the computation that follows, but these should be invoked in the order in which they are used,
otherwise the reader has to sort out for himself what is being used when (not very hard, admittedly, but
mildly irritating.)
I propose therefore writing
"Invoking [Bump] prop. 2.1.3., equation (148), and Lemma 5.7. in turn, we find:"
Line -5: Note that the notation \Lambda_r is being used here, to denote a scalar in K*;
this is not an actual clash of notation but to me it is still a bit close for comfort to the Lambda that
was used to denote an isomorphism between two spaces of harmonic modular forms, and the Lambda that
is used to describe a lattice. Can we use something more descriptive here as well?
Page 53
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Line 1: Replace the symbol "\in" by "belongs to".
line 2: H_1(A-tau) could be replaced perhaps by H_1(A_tau(C),Q) or something?
Fix up the punctutation in this line, and remove "given by" after "i.e.,"
Line 6: I am not terribly fond of using \cO' (i.e., {\cal O}')
to denote an order in M2(Z), since the letter \cO is often use to denote orders in the commutative
ring K. Would it be a problem to use M_0(N) instead (which is a notation that I rather like,
and is reasonably transparent and consistent with other notations)?
In the fourth displayed equation, we would then have
K \cap M_0(N) = \cO_c,
which seems more reasonable to me.
Two lines after, we would have to replace \cO'^\times by M_o(N)^\times, which
again seems reasonable to me (but leads me to a question: do we not want things to be set up in such a way
that the ambiguity is by Gamma_0(N) rather than M_0(N)^\times, by fixing some orientation at infinity
perhaps)?
Second line after 4th displayed equation, I would display the map of algebraic groups, and replace K*
by Res^K_Q G_m, even though I know that identifying the group with its Q-points is a fairly common
abuse of notation...
Line 17: This equation is a bit longish and I therefore think it should be displayed.
Likewise, two lines later, the whole part between "write" and "and set" could be displayed,
with space added between the various parts after the "with".
Proof of Lemma 5.10. The first sentence is unpleasantly terse. Write instead:
"For any prime q\nmid N, the restriction of $\omega_f$ to U_q' is trivial, and..."
Suppose therefore that q divides N. "
(Again, that Serre Rule, | should not be used as a main verb..."
Statement of Proposition 5.11. As far as I could tell this is the first time that eta and eta' are defined,
although they are used before. You should move the definition outside proposition 5.11 and display it prominently
(in a numbered equation perhaps) which you can then refer to in the statement of Prop 5.11 if the reader's memory
needs to be refreshed.
Page 54
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Line 1: "without loss.." of what? (Intelligibility, referee's good will, generality...?)
line 8: This is somewhat cramped, so you should display the equation after "we may write",
inserting spaces after the comma.
In fact this sentence is a good illustration of the necessity of Serre's Golden Rule about not seperating
mathematical clauses by commas. A reader (particularly one, like me, or even worse, Massimo,
who is getting on in years and not wearing his reading glasses) might find it hard to distinguish between your
"xi_A(y_i) = g_i(g_U,i . gamma_i), g_i \in ..."
and
"xi_A(y_i) = g_i(g_U,i . gamma_i) . g_i \in ..."
implying that xi_A is defined as a product of g_i(g_U_i.gamma_i) with g_i, and belongs to GL_{2,Q}.
This problem disappears when the entire mess is displayed and the various clauses
are clearly seperated by spaces (I think \quad is the right amount of space to insert...)
Right after that equation, I would write the sentence (again, to relieve the cramping)
"Further, since \xi is a Heegner embedding, the element g_i g_{U,i} belongs to \hat {\cal O}'".
(The latter symbol, if you take my earlier advice about \cO' to heart, would become
\hat M_0(N)^\times, which I find is not too heavy and more descriptive...)
Equation (155), insert a \quad of space before c_i
Displayed equation -5 on the page: Put parentheses around
Z(\gamma_i\tau) + Z
Page 55
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Line 8: Write
"On SL)2(A), *the representation* r_psi is defined on the matrices
$$ *display the matrices U(a) d(a) and W as per the displayed equation below * $$
by the formulas:
$$ insert your three formulas, taking care to correct the punctuation, $$
where (here there are three things to define, so I propose enumerating them list-style, with bullets):
* (.,.)_v denotes the Hilbert symbol, so that ... associated to V,
* gamma_v is an eigth root ...
* phi-hat denotes the fourier transform, defined by...
Page 56
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Definition ofthe Schwartz function after Theorem 5.12,
(i): mention that \varphi_q is the same here as \varphi_q^\Xi.
Displayed equation before Lemma 5.14: Insert space before a,d, b, and c.
Page 57
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Second displayed equation: these matrices were aleady defined in displayed form on page 55, albeit in a
different order and with a instead of y for U(y).
I don't object strongly to redisplaying the information, but it gives a feeling of something
hastily written.
In the middle of the page, perhaps the definition of V(z), which I think appears for the first time
and is used later, deserves on the other hand to
be displayed.
Page 58
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Be careful with punctuation particularly around displayed formulas. For instance there should be a comma
at the end of (159) and a period in the penultimate and -4th displayed equations.
Last line of Page 58: you should state what is this "Casselman's Theorem",
or perhaps just provde a precise reference. It is probably well-known to the experts
but not to me (and, I gathered, not to Massimo either...)
Also, presumably Casselman has more than one theorem so this might be a bit ambiguous!
Page 59
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Line 2: insert "for" after V(z).
Also, perhaps we want to remind the reader of where V(z) is defined (this is less important if the
definition is made into a displayed equation in any case.)
line 3: 'transforms by ... *under* the right action of D(a,b)."
First displayed equation, put in some spaces as before.
Line -11: This sequence of two equalities looks a bit cramped and could be displayed.
Right after that, the matrix which you re-define as w was called W on page 57;
the sentence would be clearer if you specifty that x now ranges over F_q.
Last line before Def. 5.16, does loc. cit. refer to [SR], Proposition 2.12? If so, it is better
to rewrite the reference since it occured a while back, and if no, a precise reference should be entered.
Page 60
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In the last paragraph, in the process of checking the compatibility between what you write and Prop. 4.3.4.. of Xue,
it would be helpful to the reader to include a little explanation of how our notations differ from those of
Xue in the statement of his result.
Page 61
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Line 8: I propose diplaying the longish equation that comes after "But"
Definition 5.19: The statement is a bit awkward: perhaps write "The explicit Schwartz function \varphi is defined as...
and then carry on as before.
Section 5.5. second line: I propose to display the parts after "Then" and insert spaces as needed.
Lemma 5.12: It is in an equation of this sort that one sees why it is better to use
{\cal H } and not a regular H for the upper half plane, since H is now a group...
Also, perhaps the restriction to H(A) is not necessary to indicate so explicitly in the integrand, since your
notation already indicates that you are integrating over H(Q) \ H(A).
(removing this would ligten the expression somewhat.)
Page 62
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Line 1: Add space, and display perhaps to make more readable.
Page 63
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Take care of punctuation after first and second displayed equations.