Subject:1chapters 1,2 of your thesis2Date:3Sun, 26 Mar 2000 11:53:30 -0800 (PST)4From:5Bjorn Poonen <[email protected]>6To:7[email protected]89101112Dear 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--Bjorn24252627p.2, first sentence: if I personally were asked to name the main28outstanding problem in the arithmetic of elliptic curves, I would say29it is the problem of whether there is an algorithm to compute30Mordell-Weil ranks (or equivalently, via descent, the problem of determining31whether a genus 1 curve over a global field has a rational point).32Of course this is related to BSD, and in particular is implied by33the finiteness of Sha, but to me the latter problems are secondary.3435p.2, Def 1.1: do you want to require that A be simple over Q?36(It's up to you.)3738p.3, line 7: longterm should be long-term3940p.3, Cor 1.4: you could replace "is an integer, up to a unit in"41by $\in$4243p.4, line -3 (i.e., 3 lines from the bottom): "1-dimensional abelian44varieties": why not call them elliptic curves4546p.5, Thm 1.7: define $\rho_{E,p}$4748p.5, line -3: "one expects..." Is there some theoretical heuristic49for this? If not, it might be more accurate to write50"numerical experiments suggest..."51Also (to be picky), when you write "most of III(A-dual)"52you don't really mean most of III(A-dual) for each A,53but most as you VARY A, I am guessing.5455p.6: "So far there is absolutely no evidence..."56I guess there is no evidence to lead one to conjecture the opposite, either.57I guess I don't understand your reasons for writing this sentence.58By writing it this way, do you mean to suggest that you are more59inclined to believe that III(A-dual) is eventually all visible60in some J_0(NM)?6162p.6, next paragraph: significant difficult6364p.6, first sentence of 1.1.6: "...is bound to fail."65This sounds as if you've proved that it will fail.66If you haven't, maybe it would be better to say "will probably fail".6768p.6, 1.1.6: give a reference for Kani's conjecture.6970p.6, first sentence of 1.2: remove the comma7172p.6, second sentence of 1.2: "to provably compute"73(grammatically speaking, it's incorrect to split an infinitive)7475p.6, Theorem 1.8: The sentence beginning "Suppose p is an odd prime..."76sounds a little funny to my ear. I'd suggest replacing the "and" by77"or the order"7879p.6, last "sentence" of Theorem 1.8: It's better to avoid using80a symbol as the verb of a sentence. You could instead say81"Then there exists an injection82B(Q)/p...."8384p.7, a little over halfway down the page:85I'd suggest putting a period after "K_1=0" and then beginning a new sentence.8687p.7, next paragraph: "the latter group contains infinitely many elements88of order p" Maybe give a reference for this? (even though you don't89use it)9091p.7, next paragraph: archimedean9293p.7, next paragraph: define Q_v^ur. Also, give a reference for the94generalization of Tate uniformization.9596p.7, same paragraph: "it follows that there is a point Q..."97Am I missing something?98It seems to me that this doesn't work when v=p.99I'm worried...100101p.8: when you take stalks the J suddenly becomes a B! (twice)102103p.8, middle: need a period at the end of the paragraph104105p.8, next paragaph: "The 2-primary subgroup $\Phi$ of $A \cap B$106is rational over $\Q$."107I don't see why the points in $\Phi$ have to be rational.108Oh, do you mean simply that it is rational as a subgroup?109110p.8, penultimate sentence of the proof: "the component group...has order111a power of 2". In fact, it's trivial, since A-tilde has good reduction at 2.112113p.9, line 6: quotient needs an s114115p.9, two paragraphs later: "By definition, there must be other subvarieties..."116By definition of what?117118p.9, end of that paragraph: "can not" should be "cannot" I think.119120p.10, section title of 1.3.1: move "only" after "considering"121(only should be put as close as possible to the thing it is onlifying,122if you know what I mean)123124p.15, section 2.1: Perhaps explain the motivation for these125definitions. (You probably have more intuition and knowledge about126this than I do. Is it that {a,b} was originally thought of as the127homotopy type of a path from a to b through the upper half plane (or128its projection in a modular curve). This would explain the relations,129for instance.)130131p.16, line 2: "torsion-free quotient": Are you claiming that this132quotient is already torsion-free, or that you are going to make it133torsion-free by dividing out its torsion subgroup if necessary?134If the latter, I think it'd be worth defining the term135"torsion-free quotient" separately.136137p.16, line 2: You never defined Z[epsilon].138Is it the subring of the space of functions from (Z/NZ)* to C139generated by epsilon, or the subring of C generated by the values140of epsilon, or perhaps the group ring Z[G] where G is the group141generated by epsilon? (I'm pretty sure I know the answer, but142another reader might not.)143144p.16, definition of M_k(N,epsilon;R): I think you mean145"tensor over Z[epsilon]". "Tensor" by itself means "tensor over Z,"146which will give something very different.147148p.17, first two sentences of 2.4.1: This is a little vague (and awkward).149Maybe write instead:150Let $V$ denote either a space $M_k(N,\epsilon;R)$ of modular symbols151or a space \dots of modular forms [you should clarify what sort of152spaces of forms you will consider].153The Hecke algebra $\T$ is then the subring of $\End_R(V)$154generated by the $T_n$.155156Clearly T depends on the choice of N,epsilon,R.157But given this data, is it the same for modular symbols158and for modular forms? I suppose the answer might depend on159exactly which type of modular forms you consider.160161Is it obvious what the action on antiholomorphic forms is?162163p.17, Prop 2.7: Give a reference for this, if you're not going to prove it.164165p.18, line 8: it's should be its166167p.18, Definition 2.10: since "plus one" is acting as an adjective,168I think it'd be better to put a hyphen in the middle.169Same for minus-one.170171p.20, Definition 2.14: I have some questions for you: are the new and old172modular symbols disjoint? Is their sum equal to the whole space,173or at least is their sum of finite index in the whole space?174175p.20, Remark 2.15: "can not" should be one word I think.176177p.20, Remark 2.16: Is p prime to MN? What is F_p[epsilon]?178Is it Z[epsilon]/(p), or Z[epsilon]/(fancyp) where fancyp179is a prime of Z[epsilon] above p, or ... ?180181p.20, line 4 of Remark 2.16: basis should be bases182183p.20, matrix in Remark 2.16:184Are you sure you want to write it in this transposed way?185It is much more common to write linear transformations186as matrices acting on the left on column vectors.187(If you are going to keep it as is, it might help to remark188that you are doing things this way.)189190p.20, bottom: It'd be better to define P^1(t) in a separate sentence.191When I first read this, I didn't realize that this was supposed to192be a definition of P^1(t) and I started looking back at earlier193pages searching for one.194195p.21, top: In some sense, deterministic algorithms have a greater right196to be called algorithms than random algorithms. Although I am sure197that from the implementation point of view it was easier to do things198the way you did them, you might at least add a comment that it is199possible to rewrite this a deterministic algorithm, say by first computing200coset representatives for Gamma(MN) in Gamma(1),...201202p.21, 2.5.1: "base field"? There has been no mention of base field203up to now, in the context of modular symbols.204Do you mean that you are now taking R to be a field?205By "degeneracy maps" do you mean alpha_t and beta_t206relativized to R?207208p.22, first line of proof of Theorem 2.19: tensor over Z[epsilon] again?209210p.26, middle: exists should be exist211212p.27, last line of proof of Prop 2.28: "torsion free" should be torsion-free213214215Subject:216more comments217Date:218Mon, 27 Mar 2000 23:27:05 -0800 (PST)219From:220Bjorn Poonen <[email protected]>221To:222[email protected]223224225226227Dear William:228229I've finished "reading" your thesis.230Below are the rest of my comments.231232--Bjorn233234235236p.28, diagram 2.1, etc.: I'm really puzzled by this and your237comment on p.44 that the degree of the composition238theta_f : A_f-wedge --> A_f239need not be a square. There's no contradiction, but there's240a natural approach to try to prove that it IS a square,241and I'm wondering where it goes wrong. So here are some questions242about the situation:2431) For k>2 is J_k(N,epsilon) an abelian variety?244(It was unclear to me from your remark about Shimura at the beginning245of section 2.7 whether Shimura proved this in general or not.)2462) If so, is it a PPAV ? I think this is equivalent247to the complex torus being isomorphic to its dual.2483) If a complex torus is a quotient of an abelian variety over C,249is it automatically an abelian variety? (I think yes.)2504) Is A_f-wedge --> J_k(N,epsilon) the map dual to J_k(N,epsilon) --> A_f ?2515) Is theta_f always an isogeny?252253254p.30, bottom: what does it mean to compute an O-module.255I guess what I'm really asking is, how will you present the answer?256Will you give a Z-basis?257258p.31, 2nd paragraph of 3.2: In the definition of M_k(Gamma) are you259working over C?260261p.32, line 5: "Put R=F_p in Prop 3.6" -- but just before Prop 3.6262you said that R was going to be a subring of C.263264p.32, paragraph following Lemma 3.11: a_i is an element of what?265the positive integers?266267p.32, end of this paragraph: "We thus represent epsilon as a matrix"268Why call it a matrix, if it's really just a vector?269270p.32, bottom: the ' in n'th looks a lot like an apostrophe here.271You might try $(n')^{\text{th}}$.272I personally prefer $n^{\text{th}}$ to $n$th (so much so, that I made273a macro out of it). If \text doesn't work in your brand of tex,274try \operatorname in its place.275276p.33, line before definition 3.13:277I don't understand the (2^{n-2}-1)/2.278Shouldn't it be 2^{n-3}, for n>=3 ?279280p.33, sums in Theorem 3.14: the size and spacing of the indices of281summation looks really weird.282283p.34: delete comma after "cumbersome"284285p.34, two lines later: "The author..." of this thesis or of [Hij64]?286287p.34, same sentence: "...has done this and found..." The tenses don't288match. How about "has done this and has found..."289290p.34, line -4: what is S?291292p.35, line 2 of 3.6.1: something's messed up293294p.35, (3.1): this is a little weird in that M_k(N,epsilon)295is not a K-vector space296297p.35, two lines later: how do view the elements of T298as "sitting inside M_k(N,epsilon)"?299300p.35, prop 3.15: Probably you should go back to Def 2.14301and do it over other bases, since I think here you want new302modular symbols over K, in order to get a good notion of irreducible.303304p.35, next paragraph: "The new and old subspace of M_k(N,epsilon)^perp305are defined as in Definition 2.14."306Will the alpha_t and beta_t be replaced by beta_t^perp and alpha_t^perp,307respectively?308If so, it might be worth giving the definition in full here309rather than refer back to Def 2.14.310311p.35, algorithm 3.16, lines 4-5: "Using the Hecke operators..."312Although there's nothing technically wrong with this sentence,313it tricked me into thinking it was going to be parsed differently,314if you know what I mean. Is there some way you could rewrite it?315316p.35, algorithm 3.16, 3(b): is this stated correctly?317Give a reference for the facts you are assuming, or prove them.318319p.36, top: "repeat step 1"; do you mean just step 1, or do you320mean go back to step 1. also, did you mean to replace p by321the next larger prime?322323All over the place: Some editors consider contractions (like don't)324too informal for published math.325326p.37, alg 3.19: "Then for any randomly chosen..."327What is the mathematical meaning you have in mind here?328By the way, is K infinite?329330p.37, same sentence: by my convention, g(A)v is always an eigenvector;331the real question is whether it is nonzero!332333p.38, alg 3.20, step 2: by Hecke operator, do you mean a T_n,334or any linear combination? If the former, it's not obvious that335the primitive element theorem is enough.336337p.38, step 4: "w is a freely generating Manin symbols".338Even without the "s" I'm not sure what this means.339340p.38, line -5: define K[f].341342p.38, bottom: Is it clear that these traces determine f uniquely?343344p.39, end of 3.6: is it clear that all ties will eventually be broken?345346p.41, just before def 3.25: "...we use it to computing..."347348p.42, line 5: "The rank of a square matrix equals the rank of its transpose..."349This holds even if the matrix is not square!350351p.42, first line of proof of 3.29: define O-lattice.352In particular, make clear that you insist on finite covolume.353354p.42, sentence above alg 3.30: J(Q) has not been defined.355Do you mean to say that when k=2, epsilon=1,356then J can be identified with J_0(N)(C)?357358p.43, 3.9.1: Do you know about Glenn Steven's book,359called "Arithmetic on modular curves" or something like that?360I think maybe he works out in general over which fields361cusps on modular curves are defined. This together with362modular symbol calculations should give a reasonable solution363to the problem. I'm not saying that you should carry this out;364but if you feel that Steven's book is relevant, maybe you could365cite it.366367p.43, proof of prop 3.32:368In what space are T_p and Frob+Ver equal?369Define g.370Does all this work even in the bad reduction case?371Give reference for f(t)=x^{-g} F(x), or explain.372373p.43, bottom: Are you claiming that you have a counterexample374in the form A_f ?375(By the way, you should use ; or : or . instead of ,376in the middle of this sentence.)377378p.44: give a reference for Prop 3.35379380p.44, alg 3.36: define "modular kernel"381382p.46: is it known that c_A is a positive integer?383384p.47, second line of proof of 3.4: "of" after "smooth locus"385386p.47, (3.2): give a reference for the isomorphism in the middle387388p.47, middle: should Tor^1 be Tor_1 ?389390p.47: "torsion free" should be hyphenated I think (several times)391392p.47, line -7: Is fancyB a Neron model too?393394p.47, same line: "In particular,..." How does this follow from395the exact sequence from Mazur?396397p.47, next line: why is the map on the right an isomorphism?398399p.48, line 6: singe (I don't think that's the word you want!)400401p.48, remark 3.44: peak (wrong word, again)402403p.49, top: you define g but never use it!404405p.52, middle: "It would be interesting to know whether..."406Since you then give a counterexample, maybe it would be better407to replace "whether" by "under what circumstances".408Also, instead of saying "When k is odd this is clearly not the case"409it would be better to say "This sometimes fails for odd k"410since for some odd k and certain N it will be true (for instance411when S_k(N,epsilon) is trivial!)412413p.52, line 2 of 3.13.3: "Section [AL70]" Is this a typo?414415p.52, next line: missing >416417p.53: k-2th: put k-2 in parentheses418419p.54, top: "to efficiently compute" split infinitive420421p.54, line -9: how does e_i depend on i?422423p.55, def 3.50: "time" should be "times"424425p.55, def 3.50: you shouldn't call it a -1 eigenspace,426since A_f(C) is not a vector space427428p.58, CM elliptic curves: "Let be a rational newform with complex429multiplication." What does this mean? Give a definition or a reference.430431p.60, line -6: , after "purely toric" should be .432433p.60, line -4: is A' the dual of A (which you later call A-wedge)?434435p.61, middle: on the right of the "dualize" should C be C-dual?436437p.62, middle: the T and U are backwards in the vertical sequence.438Think of the semidirect product of G_m by G_a (the "ax+b" group).439440p.62, two lines later: remove the , after "purely toric reduction"441442p.62, definition of X_A: should take Homs over the algebraic closure,443or else define X_A as a group scheme.444(For example, if T is a nontrivial twist of G_m, then Hom(T,G_m)=0,445which is not what you want.)446447p.62, sequence just before 4.3: give a reference448449p.62, thm 4.2: define universal covering450451p.64, middle: one-motif !!!452453p.65, example 4.7: "... is a Tate curve" over Q_p^ur.454(For a ramified extension, the answers will be different.)455456p.65, middle: define pi_*, pi^*, theta_*, theta^*457458p.65, bottom: prove or give a reference for the middle equality459460p.66, middle: "Suppose L is of finite index in fancyL."461This makes it sound as if L is some previously defined object.462How about replacing this by "For L of finite index in fancyL, define..."463464p.68, line 10: change "act" to "acts"465466p.68, middle: "...is a purely toric optimal quotient..."467It'd be nice to specify that this is "purely toric at p".468469p.69, end of WARNING: 3 does not make sense, since the group470has not been identified with Z/42Z.471Anyway it's probably safe to leave this out,472since people reading this will presumably know473what an order 14 subgroup of a cyclic group of order 42 looks like.474475p.69, line -3: remove ( to the right of the rightarrow476477p.70, conj 4.18: I don't see how #A_i(Q) = #Phi_{A_i} could possibly hold,478given that the former can be infinite, for instance when p=37.479480p.72, Table 4.3: where's 67 ???481482483484OK, I'm done (except for section 3.11 on which you wrote485"This section has been rewritten").486487488489490491