Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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These are Bjorn's comments on my thesis.
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"A:" or "S:"
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"S:" means I'm stumped at present (possibly because of lack of net access)
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P: is a pointer to where I'm at.
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Date: Sun, 26 Mar 2000 11:53:30 -0800 (PST)
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Subject: chapters 1,2 of your thesis
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Dear William:
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So far I've read through the end of Chapter 2 in your thesis.
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It's really very well written. I must say, however, that the
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technical nature of Chapter 2 made me want to skim through it
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rather than read every detail; I suppose that's inevitable.
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Anyway, here are the comments I have so far. You can choose to
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ignore most of them if you want; there are very few that are substantial.
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I hope you won't be offended if I sometimes complain about grammar!
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--Bjorn
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p.2, first sentence: if I personally were asked to name the main
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outstanding problem in the arithmetic of elliptic curves, I would say
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it is the problem of whether there is an algorithm to compute
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Mordell-Weil ranks (or equivalently, via descent, the problem of determining
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whether a genus 1 curve over a global field has a rational point).
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Of course this is related to BSD, and in particular is implied by
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the finiteness of Sha, but to me the latter problems are secondary.
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A: I changed the wording slightly, and added references to two papers
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that agree with my opinion.
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p.2, Def 1.1: do you want to require that A be simple over Q? (It's
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up to you.)
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A: No; I'll say "simple" when needed.
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p.3, line 7: longterm should be long-term
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A: OK.
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p.3, Cor 1.4: you could replace "is an integer, up to a unit in"
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by $\in$
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A: OK.
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p.4, line -3 (i.e., 3 lines from the bottom): "1-dimensional abelian
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varieties": why not call them elliptic curves
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A: OK.
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p.5, Thm 1.7: define $\rho_{E,p}$
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A: OK.
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p.5, line -3: "one expects..." Is there some theoretical heuristic
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for this?
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A: "Mazur told me so", but didn't really explain it sufficiently.
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He said something about the modular degree annihilating the
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"symmetric square", and the symmetric square should have nothing
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a priori, to do with Sha.
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If not, it might be more accurate to write
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"numerical experiments suggest..."
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Also (to be picky), when you write "most of III(A-dual)"
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you don't really mean most of III(A-dual) for each A,
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but most as you VARY A, I am guessing.
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A: Right -- thanks. I changed it to say that
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"Numerical experiment suggests that
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as $\Adual$ varies, Sha is often not visible inside
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J_0(N). For example ..."
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p.6: "So far there is absolutely no evidence..."
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I guess there is no evidence to lead one to conjecture the opposite, either.
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I guess I don't understand your reasons for writing this sentence.
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By writing it this way, do you mean to suggest that you are more
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inclined to believe that III(A-dual) is eventually all visible
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in some J_0(NM)?
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A: I re-worded it to say:
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"We have been unable to compute any examples in which $\Sha(\Adual)$ is
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not visible at level~$N$, but becomes visible at some level $NM$.
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Any data along these lines would be very interesting."
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p.6, next paragraph: significant difficult
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A: I must have deleted this or fixed it, as I can't find it.
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p.6, first sentence of 1.1.6: "...is bound to fail."
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This sounds as if you've proved that it will fail.
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If you haven't, maybe it would be better to say "will probably fail".
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A: Thanks.
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p.6, 1.1.6: give a reference for Kani's conjecture.
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--> See page 9 of Cremona-Mazur: "Kani-Shantz".
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p.6, first sentence of 1.2: remove the comma
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A: ok.
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p.6, second sentence of 1.2: "to provably compute"
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(grammatically speaking, it's incorrect to split an infinitive)
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A: ok. Now the paragraph reads:
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Without relying on any unverified conjectures,
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we will use the following theorem to exhibit abelian varieties~$A$
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such that the visible part of $\Sha(A)$ is nonzero.
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p.6, Theorem 1.8: The sentence beginning "Suppose p is an odd prime..."
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sounds a little funny to my ear. I'd suggest replacing the "and" by
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"or the order"
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A: done.
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p.6, last "sentence" of Theorem 1.8: It's better to avoid using
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a symbol as the verb of a sentence. You could instead say
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"Then there exists an injection
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B(Q)/p...."
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A: good point.
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p.7, a little over halfway down the page:
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I'd suggest putting a period after "K_1=0" and then beginning a new sentence.
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A: ok.
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p.7, next paragraph: "the latter group contains infinitely many elements
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of order p" Maybe give a reference for this? (even though you don't
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use it)
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A: ok, I'll mention Shafarevich's paper.
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p.7, next paragraph: archimedean
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A: ok
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p.7, next paragraph: define Q_v^ur. Also, give a reference for the
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generalization of Tate uniformization.
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S: look at what Ken does... Faltings-Chai??
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p.7, same paragraph: "it follows that there is a point Q..."
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Am I missing something?
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It seems to me that this doesn't work when v=p.
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I'm worried...
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A: It doesn't; I've weekend the statement of the theorem accordingly.
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I wonder how to generalize the result to include this case...?
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p.8: when you take stalks the J suddenly becomes a B! (twice)
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A: woops!
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p.8, middle: need a period at the end of the paragraph
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A: ok
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p.8, next paragaph: "The 2-primary subgroup $\Phi$ of $A \cap B$
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is rational over $\Q$."
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I don't see why the points in $\Phi$ have to be rational.
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Oh, do you mean simply that it is rational as a subgroup?
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A: I mean "rational as a subgroup", which I'll write more explicitly.
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p.8, penultimate sentence of the proof: "the component group...has order
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a power of 2". In fact, it's trivial, since A-tilde has good reduction at 2.
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A: Thanks for pointing this out.
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p.9, line 6: quotient needs an s
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A: Thanks; also, I just realized that "rank zero optimal quotients"
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should be "rank-zero optimal quotients".
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p.9, two paragraphs later: "By definition, there must be other subvarieties..."
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By definition of what?
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A: modular degree -- I reworded this paragraph more clearly.
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p.9, end of that paragraph: "can not" should be "cannot" I think.
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A: I agree.
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p.10, section title of 1.3.1: move "only" after "considering"
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(only should be put as close as possible to the thing it is onlifying,
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if you know what I mean)
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A: Thanks for the tip!
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p.15, section 2.1: Perhaps explain the motivation for these
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definitions. (You probably have more intuition and knowledge about
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this than I do. Is it that {a,b} was originally thought of as the
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homotopy type of a path from a to b through the upper half plane (or
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its projection in a modular curve). This would explain the relations,
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for instance.)
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p.16, line 2: "torsion-free quotient": Are you claiming that this
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quotient is already torsion-free, or that you are going to make it
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torsion-free by dividing out its torsion subgroup if necessary?
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If the latter, I think it'd be worth defining the term
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"torsion-free quotient" separately.
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A: I've systematically written "the largest torsion-free quotient of" in place
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of "torsion-free quotient."
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p.16, line 2: You never defined Z[epsilon].
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Is it the subring of the space of functions from (Z/NZ)* to C
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generated by epsilon, or the subring of C generated by the values
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of epsilon, or perhaps the group ring Z[G] where G is the group
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generated by epsilon? (I'm pretty sure I know the answer, but
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A: Thanks for pointing out the ambiguity.
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It's the subring of C generated by the values of eps.
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p.16, definition of M_k(N,epsilon;R): I think you mean
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"tensor over Z[epsilon]". "Tensor" by itself means "tensor over Z,"
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which will give something very different.
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A: Yep. Thanks.
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p.17, first two sentences of 2.4.1: This is a little vague (and awkward).
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Let $V$ denote either a space $M_k(N,\epsilon;R)$ of modular symbols
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or a space \dots of modular forms [you should clarify what sort of
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spaces of forms you will consider].
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The Hecke algebra $\T$ is then the subring of $\End_R(V)$
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generated by the $T_n$.
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Clearly T depends on the choice of N,epsilon,R.
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But given this data, is it the same for modular symbols
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and for modular forms? I suppose the answer might depend on
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exactly which type of modular forms you consider.
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Is it obvious what the action on antiholomorphic forms is?
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A: Ok.
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p.17, Prop 2.7: Give a reference for this, if you're not going to prove it.
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A: Ok.
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p.18, line 8: it's should be its
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A: Ok.
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p.18, Definition 2.10: since "plus one" is acting as an adjective,
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I think it'd be better to put a hyphen in the middle.
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Same for minus-one.
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A: Ok.
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p.20, Definition 2.14: I have some questions for you: are the new and old
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modular symbols disjoint? Is their sum equal to the whole space,
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or at least is their sum of finite index in the whole space?
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S: I'm not sure about the sum on the Eisenstein part
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p.20, Remark 2.15: "can not" should be one word I think.
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p.20, Remark 2.16: Is p prime to MN? What is F_p[epsilon]?
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Is it Z[epsilon]/(p), or Z[epsilon]/(fancyp) where fancyp
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is a prime of Z[epsilon] above p, or ... ?
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p.20, line 4 of Remark 2.16: basis should be bases
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p.20, matrix in Remark 2.16:
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Are you sure you want to write it in this transposed way?
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It is much more common to write linear transformations
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as matrices acting on the left on column vectors.
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(If you are going to keep it as is, it might help to remark
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that you are doing things this way.)
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p.20, bottom: It'd be better to define P^1(t) in a separate sentence.
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When I first read this, I didn't realize that this was supposed to
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be a definition of P^1(t) and I started looking back at earlier
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pages searching for one.
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p.21, top: In some sense, deterministic algorithms have a greater right
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to be called algorithms than random algorithms. Although I am sure
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that from the implementation point of view it was easier to do things
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the way you did them, you might at least add a comment that it is
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possible to rewrite this a deterministic algorithm, say by first computing
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coset representatives for Gamma(MN) in Gamma(1),...
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p.21, 2.5.1: "base field"? There has been no mention of base field
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up to now, in the context of modular symbols.
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Do you mean that you are now taking R to be a field?
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By "degeneracy maps" do you mean alpha_t and beta_t
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relativized to R?
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p.22, first line of proof of Theorem 2.19: tensor over Z[epsilon] again?
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p.26, middle: exists should be exist
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p.27, last line of proof of Prop 2.28: "torsion free" should be torsion-free
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Date: Mon, 27 Mar 2000 23:27:05 -0800 (PST)
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Dear William:
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Below are the rest of my comments.
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--Bjorn
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p.28, diagram 2.1, etc.: I'm really puzzled by this and your
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comment on p.44 that the degree of the composition
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theta_f : A_f-wedge --> A_f
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need not be a square. There's no contradiction, but there's
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a natural approach to try to prove that it IS a square,
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and I'm wondering where it goes wrong. So here are some questions
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1) For k>2 is J_k(N,epsilon) an abelian variety?
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(It was unclear to me from your remark about Shimura at the beginning
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of section 2.7 whether Shimura proved this in general or not.)
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A: Shimura's construction is different; there's no reason why ours should agree.
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2) If so, is it a PPAV ? I think this is equivalent
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to the complex torus being isomorphic to its dual.
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A: I think so, by properties of the Peterson inner product.
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3) If a complex torus is a quotient of an abelian variety over C,
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is it automatically an abelian variety? (I think yes.)
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A: Yes, at least in this case, because it's a quotient by an ideal of
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endomorphisms.
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4) Is A_f-wedge --> J_k(N,epsilon) the map dual to J_k(N,epsilon) --> A_f ?
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A: It should be, in the Shimura setting.
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5) Is theta_f always an isogeny?
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A: I think so...
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Maybe your above remark can be used to show that my construction is definitely
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{\em not} Shimura's.
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p.30, bottom: what does it mean to compute an O-module.
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I guess what I'm really asking is, how will you present the answer?
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Will you give a Z-basis?
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A: I don't really care, because I'm not analyzing efficiency.
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The algorithm here takes "compute O-module" as a black box;
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I think it would be a mistake for me to by precise about how this part of the
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computation should go... (I've added a paranthetical remark of this nature.)
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p.31, 2nd paragraph of 3.2: In the definition of M_k(Gamma) are you
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working over C?
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A: fixed.
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p.32, line 5: "Put R=F_p in Prop 3.6" -- but just before Prop 3.6
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you said that R was going to be a subring of C.
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A: fixed.
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p.32, paragraph following Lemma 3.11: a_i is an element of what?
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the positive integers?
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A: no, of the group it generates.
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p.32, end of this paragraph: "We thus represent epsilon as a matrix"
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Why call it a matrix, if it's really just a vector?
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A: woops -- thanks
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p.32, bottom: the ' in n'th looks a lot like an apostrophe here.
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You might try $(n')^{\text{th}}$.
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I personally prefer $n^{\text{th}}$ to $n$th (so much so, that I made
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a macro out of it). If \text doesn't work in your brand of tex,
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try \operatorname in its place.
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A: Some style manuals explicitly told me not to write n^{th}... and I agree with their argument.
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I'll use paranthesis.
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p.33, line before definition 3.13:
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I don't understand the (2^{n-2}-1)/2.
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Shouldn't it be 2^{n-3}, for n>=3 ?
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A: Thanks!!!
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p.33, sums in Theorem 3.14: the size and spacing of the indices of
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summation looks really weird.
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A: OK; I changed to summing over x in (Z/NZ)^*, which is more precise and eliminates
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the need for all of the "mod"'s.
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p.34: delete comma after "cumbersome"
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A: I disagree. It is an independent clause, so it is set off in commas.
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I could delete "more cumbersome" and the sentence still makes sense.
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If I had said "Alternatively, a more cumbersome, way to ..." then I should have
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deleted the comma.
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p.34, two lines later: "The author..." of this thesis or of [Hij64]?
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A: "of this thesis". fixed.
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p.34, same sentence: "...has done this and found..." The tenses don't
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match. How about "has done this and has found..."
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A: I did it.
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p.34, line -4: what is S?
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A: Oops -- a space of cusp forms.
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p.35, line 2 of 3.6.1: something's messed up
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A: Oops -- there was an extra "\".
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p.35, (3.1): this is a little weird in that M_k(N,epsilon)
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is not a K-vector space
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A: changed to ";K"
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p.35, two lines later: how do view the elements of T
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as "sitting inside M_k(N,epsilon)"?
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A: I mean, as endomorphisms of ...
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p.35, prop 3.15: Probably you should go back to Def 2.14
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and do it over other bases, since I think here you want new
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modular symbols over K, in order to get a good notion of irreducible.
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A: ok.
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p.35, next paragraph: "The new and old subspace of M_k(N,epsilon)^perp
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are defined as in Definition 2.14."
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Will the alpha_t and beta_t be replaced by beta_t^perp and alpha_t^perp,
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respectively?
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If so, it might be worth giving the definition in full here
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rather than refer back to Def 2.14.
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p.35, algorithm 3.16, lines 4-5: "Using the Hecke operators..."
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Although there's nothing technically wrong with this sentence,
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it tricked me into thinking it was going to be parsed differently,
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if you know what I mean. Is there some way you could rewrite it?
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A: re-ordered
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P: <------------------>
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p.35, algorithm 3.16, 3(b): is this stated correctly?
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Give a reference for the facts you are assuming, or prove them.
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S: OK. This was incorrectly and awkwardly stated. I've fixed it.
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(But need to add a real reference; perhaps to Loic's paper.)
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p.36, top: "repeat step 1"; do you mean just step 1, or do you
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mean go back to step 1. also, did you mean to replace p by
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the next larger prime?
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A: Thanks; this was ridiculous imprecise as it stood!
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All over the place: Some editors consider contractions (like don't)
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too informal for published math.
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A: I just fixed it with grep.
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p.37, alg 3.19: "Then for any randomly chosen..."
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What is the mathematical meaning you have in mind here?
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A: As you point out, there's no need to have the word "random".
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By the way, is K infinite?
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A: NOPE.
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p.37, same sentence: by my convention, g(A)v is always an eigenvector;
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the real question is whether it is nonzero!
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A: Oops! Your convention is my convention, too! Thanks. I fixed it.
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p.38, alg 3.20, step 2: by Hecke operator, do you mean a T_n,
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or any linear combination? If the former, it's not obvious that
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the primitive element theorem is enough.
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A: Woops. I mean the *latter*. Not only is the former "not obvious", it isn't
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true! I found the first example in the course of my computations; it
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occurs in S_2(Gamma_0(512)). I've added a note to this effect.
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p.38, step 4: "w is a freely generating Manin symbols".
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Even without the "s" I'm not sure what this means.
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A: a symbol of the form [P,(c,d)].
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p.38, line -5: define K[f].
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A: done.
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p.38, bottom: Is it clear that these traces determine f uniquely?
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A: No, but I'm sure it's true.
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However, it is clear if we include Tr(a_n), for all n. I've
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changed to this, because it is also cleaner.
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p.39, end of 3.6: is it clear that all ties will eventually be broken?
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A: I don't know.
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p.41, just before def 3.25: "...we use it to computing..."
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A: fixed.
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p.42, line 5: "The rank of a square matrix equals the rank of its transpose..."
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This holds even if the matrix is not square!
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A: Ok!
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p.42, first line of proof of 3.29: define O-lattice.
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In particular, make clear that you insist on finite covolume.
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p.42, sentence above alg 3.30: J(Q) has not been defined.
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Do you mean to say that when k=2, epsilon=1,
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then J can be identified with J_0(N)(C)?
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A: Yep.
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p.43, 3.9.1: Do you know about Glenn Steven's book,
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called "Arithmetic on modular curves" or something like that?
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I think maybe he works out in general over which fields
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cusps on modular curves are defined. This together with
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modular symbol calculations should give a reasonable solution
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to the problem. I'm not saying that you should carry this out;
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but if you feel that Steven's book is relevant, maybe you could
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cite it.
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A: I'm citing it. I'll look into this in detail after I finish
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my thesis.
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p.43, proof of prop 3.32:
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In what space are T_p and Frob+Ver equal?
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Define g.
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Does all this work even in the bad reduction case?
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Give reference for f(t)=x^{-g} F(x), or explain.
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A: I cleaned this up a lot; strangely enough, I was giving the proof
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S: I still need to give a reference for "f(t)=x^{-g} F(x)".
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p.43, bottom: Are you claiming that you have a counterexample
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in the form A_f ?
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(By the way, you should use ; or : or . instead of ,
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in the middle of this sentence.)
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A: Yep! I should state all this for general abelian varieties
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$A$... and give a reference.
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S: give a reference; see Cassels-Flynn...
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p.44: give a reference for Prop 3.35
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A: I don't know one; though, I think it is easy to prove...
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I never use this proposition, so I'll de-proposition it, and only
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mention it in passing.
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p.44, alg 3.36: define "modular kernel"
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A: done.
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p.46: is it known that c_A is a positive integer?
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A: "YES.
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p.47, second line of proof of 3.4: "of" after "smooth locus"
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A: fixed, with ou.
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p.47, (3.2): give a reference for the isomorphism in the middle
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A: done.
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p.47, middle: should Tor^1 be Tor_1 ?
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A: done.
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p.47: "torsion free" should be hyphenated I think (several times)
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A: hmm.
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p.47, line -7: Is fancyB a Neron model too?
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...
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p.47, same line: "In particular,..." How does this follow from
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the exact sequence from Mazur?
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...
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p.47, next line: why is the map on the right an isomorphism?
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...
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p.48, line 6: singe (I don't think that's the word you want!)
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...
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p.48, remark 3.44: peak (wrong word, again)
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p.49, top: you define g but never use it!
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A: I mean "f".
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p.52, middle: "It would be interesting to know whether..."
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Since you then give a counterexample, maybe it would be better
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to replace "whether" by "under what circumstances".
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Also, instead of saying "When k is odd this is clearly not the case"
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it would be better to say "This sometimes fails for odd k"
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since for some odd k and certain N it will be true (for instance
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when S_k(N,epsilon) is trivial!)
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A: thanks!!
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p.52, line 2 of 3.13.3: "Section [AL70]" Is this a typo?
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A: oops; I used "cite" instead of "ref".
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p.52, next line: missing >
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A: got it.
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p.53: k-2th: put k-2 in parentheses
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A: Ok.
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p.54, top: "to efficiently compute" split infinitive
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A: *doh*
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p.54, line -9: how does e_i depend on i?
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A: woops.
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p.55, def 3.50: "time" should be "times"
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A: OK.
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p.55, def 3.50: you shouldn't call it a -1 eigenspace,
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since A_f(C) is not a vector space
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A: Ok.
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p.58, CM elliptic curves: "Let be a rational newform with complex
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multiplication." What does this mean? Give a definition or a reference.
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A: I'll give a silly, but correct, definition.
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p.60, line -6: , after "purely toric" should be .
576
A: thanks.
577
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p.60, line -4: is A' the dual of A (which you later call A-wedge)?
579
A: yep; you guessed it, I changed my mind at some point...
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p.61, middle: on the right of the "dualize" should C be C-dual?
582
A: yep.
583
584
p.62, middle: the T and U are backwards in the vertical sequence.
585
Think of the semidirect product of G_m by G_a (the "ax+b" group).
586
A: I disagree with you here. The torus is the connected sub-thing,
587
not the quotient.
588
589
p.62, two lines later: remove the , after "purely toric reduction"
590
A: woops.
591
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p.62, definition of X_A: should take Homs over the algebraic closure,
593
or else define X_A as a group scheme.
594
(For example, if T is a nontrivial twist of G_m, then Hom(T,G_m)=0,
595
which is not what you want.)
596
A: yep. thanks.
597
598
p.62, sequence just before 4.3: give a reference
599
A: ok.
600
S: but I should be more precise.
601
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p.62, thm 4.2: define universal covering
603
A: refered to coleman's paper where everything is defined.
604
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p.64, middle: one-motif !!!
606
A: got it.
607
608
p.65, example 4.7: "... is a Tate curve" over Q_p^ur.
609
(For a ramified extension, the answers will be different.)
610
A: ok.
611
612
p.65, middle: define pi_*, pi^*, theta_*, theta^*
613
A: ok.
614
615
p.65, bottom: prove or give a reference for the middle equality
616
S: I'll have to dig up a reference for this compatibility when I get back.
617
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p.66, middle: "Suppose L is of finite index in fancyL."
619
This makes it sound as if L is some previously defined object.
620
How about replacing this by "For L of finite index in fancyL, define..."
621
622
p.68, line 10: change "act" to "acts"
623
A: thanks.
624
625
p.68, middle: "...is a purely toric optimal quotient..."
626
It'd be nice to specify that this is "purely toric at p".
627
A: good.
628
629
p.69, end of WARNING: 3 does not make sense, since the group
630
has not been identified with Z/42Z.
631
Anyway it's probably safe to leave this out,
632
since people reading this will presumably know
633
what an order 14 subgroup of a cyclic group of order 42 looks like.
634
635
p.69, line -3: remove ( to the right of the rightarrow
636
A: got it.
637
638
p.70, conj 4.18: I don't see how #A_i(Q) = #Phi_{A_i} could possibly hold,
639
given that the former can be infinite, for instance when p=37.
640
A: I mean "A_i(Q)_tor".
641
642
p.72, Table 4.3: where's 67 ???
643
A: it's omitted.
644