These are Bjorn's comments on my thesis.1Each is answered with either2"A:" or "S:"3"A:" means the answer4"S:" means I'm stumped at present (possibly because of lack of net access)56P: is a pointer to where I'm at.78Date: Sun, 26 Mar 2000 11:53:30 -0800 (PST)9To: [email protected]10Subject: chapters 1,2 of your thesis1112Dear William:1314So far I've read through the end of Chapter 2 in your thesis.15It's really very well written. I must say, however, that the16technical nature of Chapter 2 made me want to skim through it17rather than read every detail; I suppose that's inevitable.1819Anyway, here are the comments I have so far. You can choose to20ignore most of them if you want; there are very few that are substantial.21I hope you won't be offended if I sometimes complain about grammar!2223--Bjorn2425p.2, first sentence: if I personally were asked to name the main26outstanding problem in the arithmetic of elliptic curves, I would say27it is the problem of whether there is an algorithm to compute28Mordell-Weil ranks (or equivalently, via descent, the problem of determining29whether a genus 1 curve over a global field has a rational point).30Of course this is related to BSD, and in particular is implied by31the finiteness of Sha, but to me the latter problems are secondary.3233A: I changed the wording slightly, and added references to two papers34that agree with my opinion.3536p.2, Def 1.1: do you want to require that A be simple over Q? (It's37up to you.)38A: No; I'll say "simple" when needed.3940p.3, line 7: longterm should be long-term41A: OK.4243p.3, Cor 1.4: you could replace "is an integer, up to a unit in"44by $\in$45A: OK.4647p.4, line -3 (i.e., 3 lines from the bottom): "1-dimensional abelian48varieties": why not call them elliptic curves49A: OK.5051p.5, Thm 1.7: define $\rho_{E,p}$52A: OK.5354p.5, line -3: "one expects..." Is there some theoretical heuristic55for this?56A: "Mazur told me so", but didn't really explain it sufficiently.57He said something about the modular degree annihilating the58"symmetric square", and the symmetric square should have nothing59a priori, to do with Sha.6061If not, it might be more accurate to write62"numerical experiments suggest..."63Also (to be picky), when you write "most of III(A-dual)"64you don't really mean most of III(A-dual) for each A,65but most as you VARY A, I am guessing.66A: Right -- thanks. I changed it to say that67"Numerical experiment suggests that68as $\Adual$ varies, Sha is often not visible inside69J_0(N). For example ..."7071p.6: "So far there is absolutely no evidence..."72I guess there is no evidence to lead one to conjecture the opposite, either.73I guess I don't understand your reasons for writing this sentence.74By writing it this way, do you mean to suggest that you are more75inclined to believe that III(A-dual) is eventually all visible76in some J_0(NM)?77A: I re-worded it to say:78"We have been unable to compute any examples in which $\Sha(\Adual)$ is79not visible at level~$N$, but becomes visible at some level $NM$.80Any data along these lines would be very interesting."8182p.6, next paragraph: significant difficult83A: I must have deleted this or fixed it, as I can't find it.8485p.6, first sentence of 1.1.6: "...is bound to fail."86This sounds as if you've proved that it will fail.87If you haven't, maybe it would be better to say "will probably fail".88A: Thanks.8990p.6, 1.1.6: give a reference for Kani's conjecture.91S: ask Frey? ask Ernst?92--> See page 9 of Cremona-Mazur: "Kani-Shantz".9394p.6, first sentence of 1.2: remove the comma95A: ok.9697p.6, second sentence of 1.2: "to provably compute"98(grammatically speaking, it's incorrect to split an infinitive)99A: ok. Now the paragraph reads:100Without relying on any unverified conjectures,101we will use the following theorem to exhibit abelian varieties~$A$102such that the visible part of $\Sha(A)$ is nonzero.103104p.6, Theorem 1.8: The sentence beginning "Suppose p is an odd prime..."105sounds a little funny to my ear. I'd suggest replacing the "and" by106"or the order"107A: done.108109p.6, last "sentence" of Theorem 1.8: It's better to avoid using110a symbol as the verb of a sentence. You could instead say111"Then there exists an injection112B(Q)/p...."113A: good point.114115p.7, a little over halfway down the page:116I'd suggest putting a period after "K_1=0" and then beginning a new sentence.117A: ok.118119p.7, next paragraph: "the latter group contains infinitely many elements120of order p" Maybe give a reference for this? (even though you don't121use it)122A: ok, I'll mention Shafarevich's paper.123124p.7, next paragraph: archimedean125A: ok126127p.7, next paragraph: define Q_v^ur. Also, give a reference for the128generalization of Tate uniformization.129S: look at what Ken does... Faltings-Chai??130131p.7, same paragraph: "it follows that there is a point Q..."132Am I missing something?133It seems to me that this doesn't work when v=p.134I'm worried...135A: It doesn't; I've weekend the statement of the theorem accordingly.136I wonder how to generalize the result to include this case...?137138p.8: when you take stalks the J suddenly becomes a B! (twice)139A: woops!140141p.8, middle: need a period at the end of the paragraph142A: ok143144p.8, next paragaph: "The 2-primary subgroup $\Phi$ of $A \cap B$145is rational over $\Q$."146I don't see why the points in $\Phi$ have to be rational.147Oh, do you mean simply that it is rational as a subgroup?148A: I mean "rational as a subgroup", which I'll write more explicitly.149150p.8, penultimate sentence of the proof: "the component group...has order151a power of 2". In fact, it's trivial, since A-tilde has good reduction at 2.152A: Thanks for pointing this out.153154p.9, line 6: quotient needs an s155A: Thanks; also, I just realized that "rank zero optimal quotients"156should be "rank-zero optimal quotients".157158p.9, two paragraphs later: "By definition, there must be other subvarieties..."159By definition of what?160A: modular degree -- I reworded this paragraph more clearly.161162p.9, end of that paragraph: "can not" should be "cannot" I think.163A: I agree.164165p.10, section title of 1.3.1: move "only" after "considering"166(only should be put as close as possible to the thing it is onlifying,167if you know what I mean)168A: Thanks for the tip!169170p.15, section 2.1: Perhaps explain the motivation for these171definitions. (You probably have more intuition and knowledge about172this than I do. Is it that {a,b} was originally thought of as the173homotopy type of a path from a to b through the upper half plane (or174its projection in a modular curve). This would explain the relations,175for instance.)176A: Yep; I'll add this.177178p.16, line 2: "torsion-free quotient": Are you claiming that this179quotient is already torsion-free, or that you are going to make it180torsion-free by dividing out its torsion subgroup if necessary?181If the latter, I think it'd be worth defining the term182"torsion-free quotient" separately.183A: I've systematically written "the largest torsion-free quotient of" in place184of "torsion-free quotient."185186p.16, line 2: You never defined Z[epsilon].187Is it the subring of the space of functions from (Z/NZ)* to C188generated by epsilon, or the subring of C generated by the values189of epsilon, or perhaps the group ring Z[G] where G is the group190generated by epsilon? (I'm pretty sure I know the answer, but191another reader might not.)192A: Thanks for pointing out the ambiguity.193It's the subring of C generated by the values of eps.194195p.16, definition of M_k(N,epsilon;R): I think you mean196"tensor over Z[epsilon]". "Tensor" by itself means "tensor over Z,"197which will give something very different.198A: Yep. Thanks.199200p.17, first two sentences of 2.4.1: This is a little vague (and awkward).201Maybe write instead:202Let $V$ denote either a space $M_k(N,\epsilon;R)$ of modular symbols203or a space \dots of modular forms [you should clarify what sort of204spaces of forms you will consider].205The Hecke algebra $\T$ is then the subring of $\End_R(V)$206generated by the $T_n$.207208Clearly T depends on the choice of N,epsilon,R.209But given this data, is it the same for modular symbols210and for modular forms? I suppose the answer might depend on211exactly which type of modular forms you consider.212213Is it obvious what the action on antiholomorphic forms is?214A: Ok.215216p.17, Prop 2.7: Give a reference for this, if you're not going to prove it.217A: Ok.218219p.18, line 8: it's should be its220A: Ok.221222p.18, Definition 2.10: since "plus one" is acting as an adjective,223I think it'd be better to put a hyphen in the middle.224Same for minus-one.225A: Ok.226227p.20, Definition 2.14: I have some questions for you: are the new and old228modular symbols disjoint? Is their sum equal to the whole space,229or at least is their sum of finite index in the whole space?230S: I'm not sure about the sum on the Eisenstein part231232p.20, Remark 2.15: "can not" should be one word I think.233234p.20, Remark 2.16: Is p prime to MN? What is F_p[epsilon]?235Is it Z[epsilon]/(p), or Z[epsilon]/(fancyp) where fancyp236is a prime of Z[epsilon] above p, or ... ?237238p.20, line 4 of Remark 2.16: basis should be bases239240p.20, matrix in Remark 2.16:241Are you sure you want to write it in this transposed way?242It is much more common to write linear transformations243as matrices acting on the left on column vectors.244(If you are going to keep it as is, it might help to remark245that you are doing things this way.)246247p.20, bottom: It'd be better to define P^1(t) in a separate sentence.248When I first read this, I didn't realize that this was supposed to249be a definition of P^1(t) and I started looking back at earlier250pages searching for one.251252p.21, top: In some sense, deterministic algorithms have a greater right253to be called algorithms than random algorithms. Although I am sure254that from the implementation point of view it was easier to do things255the way you did them, you might at least add a comment that it is256possible to rewrite this a deterministic algorithm, say by first computing257coset representatives for Gamma(MN) in Gamma(1),...258259p.21, 2.5.1: "base field"? There has been no mention of base field260up to now, in the context of modular symbols.261Do you mean that you are now taking R to be a field?262By "degeneracy maps" do you mean alpha_t and beta_t263relativized to R?264265p.22, first line of proof of Theorem 2.19: tensor over Z[epsilon] again?266267p.26, middle: exists should be exist268269p.27, last line of proof of Prop 2.28: "torsion free" should be torsion-free270271272Date: Mon, 27 Mar 2000 23:27:05 -0800 (PST)273To: [email protected]274Subject: more comments275276Dear William:277278I've finished "reading" your thesis.279Below are the rest of my comments.280281--Bjorn282283p.28, diagram 2.1, etc.: I'm really puzzled by this and your284comment on p.44 that the degree of the composition285theta_f : A_f-wedge --> A_f286need not be a square. There's no contradiction, but there's287a natural approach to try to prove that it IS a square,288and I'm wondering where it goes wrong. So here are some questions289about the situation:2901) For k>2 is J_k(N,epsilon) an abelian variety?291(It was unclear to me from your remark about Shimura at the beginning292of section 2.7 whether Shimura proved this in general or not.)293294A: Shimura's construction is different; there's no reason why ours should agree.2952962) If so, is it a PPAV ? I think this is equivalent297to the complex torus being isomorphic to its dual.298299A: I think so, by properties of the Peterson inner product.3003013) If a complex torus is a quotient of an abelian variety over C,302is it automatically an abelian variety? (I think yes.)303304A: Yes, at least in this case, because it's a quotient by an ideal of305endomorphisms.3063074) Is A_f-wedge --> J_k(N,epsilon) the map dual to J_k(N,epsilon) --> A_f ?308309A: It should be, in the Shimura setting.3103115) Is theta_f always an isogeny?312313A: I think so...314Maybe your above remark can be used to show that my construction is definitely315{\em not} Shimura's.316317p.30, bottom: what does it mean to compute an O-module.318I guess what I'm really asking is, how will you present the answer?319Will you give a Z-basis?320A: I don't really care, because I'm not analyzing efficiency.321The algorithm here takes "compute O-module" as a black box;322I think it would be a mistake for me to by precise about how this part of the323computation should go... (I've added a paranthetical remark of this nature.)324325p.31, 2nd paragraph of 3.2: In the definition of M_k(Gamma) are you326working over C?327A: fixed.328p.32, line 5: "Put R=F_p in Prop 3.6" -- but just before Prop 3.6329you said that R was going to be a subring of C.330A: fixed.331332p.32, paragraph following Lemma 3.11: a_i is an element of what?333the positive integers?334A: no, of the group it generates.335336p.32, end of this paragraph: "We thus represent epsilon as a matrix"337Why call it a matrix, if it's really just a vector?338A: woops -- thanks339340p.32, bottom: the ' in n'th looks a lot like an apostrophe here.341You might try $(n')^{\text{th}}$.342I personally prefer $n^{\text{th}}$ to $n$th (so much so, that I made343a macro out of it). If \text doesn't work in your brand of tex,344try \operatorname in its place.345A: Some style manuals explicitly told me not to write n^{th}... and I agree with their argument.346I'll use paranthesis.347348p.33, line before definition 3.13:349I don't understand the (2^{n-2}-1)/2.350Shouldn't it be 2^{n-3}, for n>=3 ?351A: Thanks!!!352353p.33, sums in Theorem 3.14: the size and spacing of the indices of354summation looks really weird.355A: OK; I changed to summing over x in (Z/NZ)^*, which is more precise and eliminates356the need for all of the "mod"'s.357358p.34: delete comma after "cumbersome"359A: I disagree. It is an independent clause, so it is set off in commas.360I could delete "more cumbersome" and the sentence still makes sense.361If I had said "Alternatively, a more cumbersome, way to ..." then I should have362deleted the comma.363364p.34, two lines later: "The author..." of this thesis or of [Hij64]?365A: "of this thesis". fixed.366367p.34, same sentence: "...has done this and found..." The tenses don't368match. How about "has done this and has found..."369A: I did it.370371p.34, line -4: what is S?372A: Oops -- a space of cusp forms.373374p.35, line 2 of 3.6.1: something's messed up375A: Oops -- there was an extra "\".376377p.35, (3.1): this is a little weird in that M_k(N,epsilon)378is not a K-vector space379A: changed to ";K"380381p.35, two lines later: how do view the elements of T382as "sitting inside M_k(N,epsilon)"?383A: I mean, as endomorphisms of ...384385p.35, prop 3.15: Probably you should go back to Def 2.14386and do it over other bases, since I think here you want new387modular symbols over K, in order to get a good notion of irreducible.388A: ok.389390p.35, next paragraph: "The new and old subspace of M_k(N,epsilon)^perp391are defined as in Definition 2.14."392Will the alpha_t and beta_t be replaced by beta_t^perp and alpha_t^perp,393respectively?394If so, it might be worth giving the definition in full here395rather than refer back to Def 2.14.396397p.35, algorithm 3.16, lines 4-5: "Using the Hecke operators..."398Although there's nothing technically wrong with this sentence,399it tricked me into thinking it was going to be parsed differently,400if you know what I mean. Is there some way you could rewrite it?401A: re-ordered402403P: <------------------>404405p.35, algorithm 3.16, 3(b): is this stated correctly?406Give a reference for the facts you are assuming, or prove them.407S: OK. This was incorrectly and awkwardly stated. I've fixed it.408(But need to add a real reference; perhaps to Loic's paper.)409410p.36, top: "repeat step 1"; do you mean just step 1, or do you411mean go back to step 1. also, did you mean to replace p by412the next larger prime?413A: Thanks; this was ridiculous imprecise as it stood!414415All over the place: Some editors consider contractions (like don't)416too informal for published math.417A: I just fixed it with grep.418419p.37, alg 3.19: "Then for any randomly chosen..."420What is the mathematical meaning you have in mind here?421A: As you point out, there's no need to have the word "random".422By the way, is K infinite?423A: NOPE.424425426p.37, same sentence: by my convention, g(A)v is always an eigenvector;427the real question is whether it is nonzero!428A: Oops! Your convention is my convention, too! Thanks. I fixed it.429430431p.38, alg 3.20, step 2: by Hecke operator, do you mean a T_n,432or any linear combination? If the former, it's not obvious that433the primitive element theorem is enough.434A: Woops. I mean the *latter*. Not only is the former "not obvious", it isn't435true! I found the first example in the course of my computations; it436occurs in S_2(Gamma_0(512)). I've added a note to this effect.437438439p.38, step 4: "w is a freely generating Manin symbols".440Even without the "s" I'm not sure what this means.441A: a symbol of the form [P,(c,d)].442443p.38, line -5: define K[f].444A: done.445446p.38, bottom: Is it clear that these traces determine f uniquely?447A: No, but I'm sure it's true.448However, it is clear if we include Tr(a_n), for all n. I've449changed to this, because it is also cleaner.450451p.39, end of 3.6: is it clear that all ties will eventually be broken?452A: I don't know.453454p.41, just before def 3.25: "...we use it to computing..."455A: fixed.456457p.42, line 5: "The rank of a square matrix equals the rank of its transpose..."458This holds even if the matrix is not square!459A: Ok!460461p.42, first line of proof of 3.29: define O-lattice.462In particular, make clear that you insist on finite covolume.463464p.42, sentence above alg 3.30: J(Q) has not been defined.465Do you mean to say that when k=2, epsilon=1,466then J can be identified with J_0(N)(C)?467A: Yep.468469p.43, 3.9.1: Do you know about Glenn Steven's book,470called "Arithmetic on modular curves" or something like that?471I think maybe he works out in general over which fields472cusps on modular curves are defined. This together with473modular symbol calculations should give a reasonable solution474to the problem. I'm not saying that you should carry this out;475but if you feel that Steven's book is relevant, maybe you could476cite it.477478A: I'm citing it. I'll look into this in detail after I finish479my thesis.480481p.43, proof of prop 3.32:482In what space are T_p and Frob+Ver equal?483Define g.484Does all this work even in the bad reduction case?485Give reference for f(t)=x^{-g} F(x), or explain.486A: I cleaned this up a lot; strangely enough, I was giving the proof487for J_0(N) instead of A_f!488S: I still need to give a reference for "f(t)=x^{-g} F(x)".489490p.43, bottom: Are you claiming that you have a counterexample491in the form A_f ?492(By the way, you should use ; or : or . instead of ,493in the middle of this sentence.)494A: Yep! I should state all this for general abelian varieties495$A$... and give a reference.496S: give a reference; see Cassels-Flynn...497498p.44: give a reference for Prop 3.35499A: I don't know one; though, I think it is easy to prove...500I never use this proposition, so I'll de-proposition it, and only501mention it in passing.502503p.44, alg 3.36: define "modular kernel"504A: done.505506p.46: is it known that c_A is a positive integer?507A: "YES.508509p.47, second line of proof of 3.4: "of" after "smooth locus"510A: fixed, with ou.511512p.47, (3.2): give a reference for the isomorphism in the middle513A: done.514515p.47, middle: should Tor^1 be Tor_1 ?516A: done.517518p.47: "torsion free" should be hyphenated I think (several times)519A: hmm.520521p.47, line -7: Is fancyB a Neron model too?522...523524p.47, same line: "In particular,..." How does this follow from525the exact sequence from Mazur?526...527528p.47, next line: why is the map on the right an isomorphism?529...530531p.48, line 6: singe (I don't think that's the word you want!)532...533534p.48, remark 3.44: peak (wrong word, again)535536p.49, top: you define g but never use it!537A: I mean "f".538539p.52, middle: "It would be interesting to know whether..."540Since you then give a counterexample, maybe it would be better541to replace "whether" by "under what circumstances".542Also, instead of saying "When k is odd this is clearly not the case"543it would be better to say "This sometimes fails for odd k"544since for some odd k and certain N it will be true (for instance545when S_k(N,epsilon) is trivial!)546A: thanks!!547548p.52, line 2 of 3.13.3: "Section [AL70]" Is this a typo?549A: oops; I used "cite" instead of "ref".550551p.52, next line: missing >552A: got it.553554p.53: k-2th: put k-2 in parentheses555A: Ok.556557p.54, top: "to efficiently compute" split infinitive558A: *doh*559560p.54, line -9: how does e_i depend on i?561A: woops.562563p.55, def 3.50: "time" should be "times"564A: OK.565566p.55, def 3.50: you shouldn't call it a -1 eigenspace,567since A_f(C) is not a vector space568A: Ok.569570p.58, CM elliptic curves: "Let be a rational newform with complex571multiplication." What does this mean? Give a definition or a reference.572A: I'll give a silly, but correct, definition.573574p.60, line -6: , after "purely toric" should be .575A: thanks.576577p.60, line -4: is A' the dual of A (which you later call A-wedge)?578A: yep; you guessed it, I changed my mind at some point...579580p.61, middle: on the right of the "dualize" should C be C-dual?581A: yep.582583p.62, middle: the T and U are backwards in the vertical sequence.584Think of the semidirect product of G_m by G_a (the "ax+b" group).585A: I disagree with you here. The torus is the connected sub-thing,586not the quotient.587588p.62, two lines later: remove the , after "purely toric reduction"589A: woops.590591p.62, definition of X_A: should take Homs over the algebraic closure,592or else define X_A as a group scheme.593(For example, if T is a nontrivial twist of G_m, then Hom(T,G_m)=0,594which is not what you want.)595A: yep. thanks.596597p.62, sequence just before 4.3: give a reference598A: ok.599S: but I should be more precise.600601p.62, thm 4.2: define universal covering602A: refered to coleman's paper where everything is defined.603604p.64, middle: one-motif !!!605A: got it.606607p.65, example 4.7: "... is a Tate curve" over Q_p^ur.608(For a ramified extension, the answers will be different.)609A: ok.610611p.65, middle: define pi_*, pi^*, theta_*, theta^*612A: ok.613614p.65, bottom: prove or give a reference for the middle equality615S: I'll have to dig up a reference for this compatibility when I get back.616617p.66, middle: "Suppose L is of finite index in fancyL."618This makes it sound as if L is some previously defined object.619How about replacing this by "For L of finite index in fancyL, define..."620621p.68, line 10: change "act" to "acts"622A: thanks.623624p.68, middle: "...is a purely toric optimal quotient..."625It'd be nice to specify that this is "purely toric at p".626A: good.627628p.69, end of WARNING: 3 does not make sense, since the group629has not been identified with Z/42Z.630Anyway it's probably safe to leave this out,631since people reading this will presumably know632what an order 14 subgroup of a cyclic group of order 42 looks like.633634p.69, line -3: remove ( to the right of the rightarrow635A: got it.636637p.70, conj 4.18: I don't see how #A_i(Q) = #Phi_{A_i} could possibly hold,638given that the former can be infinite, for instance when p=37.639A: I mean "A_i(Q)_tor".640641p.72, Table 4.3: where's 67 ???642A: it's omitted.643S: add it?644645OK, I'm done (except for section 3.11 on which you wrote646"This section has been rewritten").647648I'm done too.649650651652653