Dear Henri, Michael, Sam, and Victor, Instead of writing my appendix this morning, I decided to read through your paper. I had a lot of remarks, which I've listed below, since they might be helpful in you finishing the paper soon. 1. In paragraph 1 on page 1 you define P_g to be an element of A_f(Q), but elsewhere in the paper it is not. This is confusing to me. 2. SAGE |--> "Sage" everywhere. We deprecated the all caps "SAGE" several years ago. Also, please cite Sage as explained here: http://wiki.sagemath.org/Publications_using_SAGE 3. It would be nice to add a little Acknowledgement section, and for example acknowledge the Arizona Winter School, since that is supported by an NSF grant and this paper benefited from it. 4. Page 3: Specificaly --> Specifically (spelling error) 5. Page 3: "at at" --> "at" (typo) 6. Page 4: A technicality. In section 2.1 the proper inclusion of Y_0(N) in Y is only proper when N > 1. When N=1 it is not proper. 7. Page 4: Mathematical mistake? In section 2.2 you seem to implicitly assume that Manin's conjecture that the Manin constant c=c_E is 1 is True. I think this is still not known in generality. See http://wstein.org/papers/ars-manin/ for what is known. In particular, it is known when N_E <= 60000, or so, so you're in fine shape for all examples you consider. 8. Page 4: Citation. In section 2.2 it might be nice to cite Cremona's book for the computation of pi_E^{an}, since it's explained well there, I think. 9. Page 5, Definition 2.2.1. Even given the remark 2.2.2, I'm sort of annoyed about this switcheroo, in that suddenly P is no longer in E(Q), though it was in paragraph 1 of the paper. It is almost like you have replaced the optimal *quotient* A_f = E by its dual A_f^ = E subset J_0(N). I think changing the first paragraph of page 1 to agree with this definition would be a reasonable fix. 10. Page 5, Lemma 2.3.1. I think there should be a comma after "i.e.". 11. Page 5, near end of proof of Lemma 2.3.1. "the the" --> "the" (typo). 12. Page 6, beginning of Section 3. Change the latex "We retain all the notation from \S \S 1-2." to "We retain all the notation from \S \S 1--2.", i.e., use a double dash for ranges. 13. Page 7: "righthand" is not a word, but "right-hand" and "right hand" are OK. (Despite my wife, who is Native American, have the last name "Lefthand".) 14. Page 7, section 3.1: "of of" --> "of" (typo) 15. Page 8, below lema: I think "powerseries" isn't a single word. 16. Page 9, "Fixme4 at the bottom". My book has a section about generating the Hecke algebra. This is Section 9.4.2. This is better than citing what you currently call [S1], since Chapter 9 of my book has a complete self-contained proof of everything relevant to this, whereas [S1] references a paper of Sturm. Also, Chapter 9 of my book explains a trick of Buzzard that greatly improves the result. 17. The bib reference you use for my book looks wonky. Can you use this instead, which comes from MathSciNet, but with the URL http://wstein.org/books/modform/ included somewhere: @book {stein:modform, AUTHOR = {Stein, William}, TITLE = {Modular {F}orms, {A} {C}omputational {A}pproach}, SERIES = {Graduate Studies in Mathematics}, VOLUME = {79}, NOTE = {With an appendix by Paul E. Gunnells,{\mbox{}\\\url{http://wstein.org/books/modform/}}}, PUBLISHER = {American Mathematical Society}, ADDRESS = {Providence, RI}, YEAR = {2007}, PAGES = {xvi+268}, ISBN = {978-0-8218-3960-7; 0-8218-3960-8}, MRCLASS = {11-01 (11F11 11F67 11Y16)}, MRNUMBER = {2289048}, } or \bibitem[Ste07]{stein:modform} William Stein, \emph{Modular {F}orms, {A} {C}omputational {A}pproach}, Graduate Studies in Mathematics, vol.~79, American Mathematical Society, Providence, RI, 2007, With an appendix by Paul E. Gunnells,{\mbox{}\\\url{http://wstein.org/books/modform/}}. \MR{2289048} 18. Page 11, Lemma 3.4.1, The spacing of "$X$.For" is all messed up by commenting something out in the LaTeX but not including a space? 19. Page 12, Right after proof of Prop 3.4.2. You mention integrating homology classes to compute the periods Omega_f^{+/-} for some reason. However, we already know the Weierstrass equation for our curve, so it is easy to compute the periods much more quickly using the Gauss arithmetic geometry mean (AGM). 20. Page 13, section 3.5.2: It says "We remark that the interest of computing the Chow-Heegner points $P_{g,f}$ is not as a tool for computing $E(\mathbf Q)$". I would recommend a slight change, to "We remark that our interest in this paper of computing the Chow-Heegner points $P_{g,f}$ is not as a tool for computing $E(\mathbf Q)$." I suggest this small change only because I can easily imagine somebody trying with surprising success to use some method along these lines a decade from now to actually find a rational point in some tricky context where Heegner points are hard to use. You never know. 21. Page 13: "Since each eigenvalue is an algebraic integer, then the image lies" --> "Since each eigenvalue is an algebraic integer, the image lies". 22. Page 15, Section 3.7.2: I am concerned that the following statement is wrong: "By Deligne's proof of the Ramanujan-Petersson conjecture, the cusp forms have $n$th coefficient of size $O_\epsilon(n^{\frac{1}{2}+\epsilon})$." I think you get this for $n$ a prime, but $n$ composite the bound is $O_\epsilon(n^{1+\epsilon})$. Anyway, this reminds me of Lemma 2.9 of http://wstein.org/papers/bsdalg/bsdalg2.pdf 23. Page 16, Section 4.1, first numerical example. It would be nice to have a quick pointer back in the paper to how matrices correspond to elements of homology, just for those who skip to the numerical examples section. In general, at least for the 37a1 example, it would be nice to have a whole bunch of references to sections 2--3 of the paper, instead of no references, like you currently have. 24. Page 16, Section 4.1. I would change "labeled '37a1' in..." to "labeled {\bf 37a1} in..." since Barry Mazur established that bold convention for how to typeset Cremona labels in text, and many of us have been following it. 25. Page 17. I would delete "laptop computer", since these days a lot of laptop computers (like mine) are *faster* than a typical high end server when running a single process. Servers are only better these days typically when running a highly parallel computation. It would be better to emphasize that you didn't do anything to parallelize the code (or just delete the words "laptop computer" and leave it at that). Laptops were noticeably slower than desktops until recently, but now I think putting this gives no useful information. 26. Page 17, "Thus the point we have denoted $P_{g,f}$ is the wrong thing"... This needs to be clarified. Again, this is a problem with the two competing definitions of P_{g,f} you have: one in paragraph 1 on page 1, and the other a few pages later. And I think the competition is between E viewed as a subvariety of J_0(N), which you find computationally more pleasant, and E viewed as a quotient (which I find more computationally pleasant for my appendix, by the way). In my experience, this tension permeates explicit computation with modular abelian varieties. 27. Page 18, "In table 1" should be "In Table 1". 28. Page 18, "differnetials" --> "differentials" (typo) 29. Page 18. To make this more timeless, and in case Sage changes, or somebody wants to do this stuff without Sage, the ordering isn't mysterious. The ordering is "lexicographically by the sequence of traces of Hecke operators $T_p$, for all primes $p$." That's it. It's really, really simple. Incidentally, since "Trace(T_1) = dimension", this first orders by the dimension. 30. Page 18. Regarding - "The points $P_{g,f}$ are listed as multiples of a generator $P$ for $E(\mathbf Q)$ computed using {\tt SAGE}." Sadly, this isn't well defined, since Sage just reports the generators that Cremona's mwrank program computes, and they are random. Of course, the multiple is well defined up to sign, so long as you work modulo torsion. An easy way out of this problem would be to either explicitly give the choice of generator of E(Q) that you used to make the table in the table, or instead report the absolute value of the multiple, which is less information. Sorry that this isn't necessarily well defined, but I believe mwrank can give different but equivalent output for the same input... or at least, it can change between versions of Sage/mwrank. It would be useful, by the way, to say which version of Sage you used (that should be in the bibtex ref you use for Sage). 31. Page 18. "Allow f to be an oldform." I should likely be able to do an example of this in my table, using "Zhang's description". 32. Page 18: "p-adic Rankin L-series" -- in the first ref the p and L should be in math mode and aren't. In the second ref p is in math mode, but L is not. 33. Page 20. "255-282" should be "255--282". 34. Page 20, The reference for Csirik et al. [CWZ] is incomplete. 35. Page 20, Reference for [G] is incomplete. 36. Page 20, Reference for [R] has a typo: "ivnariant". 37. Page 20, Reference for [S1] looks pretty incomplete/ugly. This is probably better: @Article{ generatinghecke, author = "Agashe, A. and Stein, W.\thinspace{}A.", title = {Generating the \protect{H}ecke algebra (appendix to {L}ario-{S}choof)}, JOURNAL = {Experiment. Math.}, FJOURNAL = {Experimental Mathematics}, VOLUME = {11}, YEAR = {2002}, NUMBER = {2}, PAGES = {303--311}, ISSN = {1058-6458}, note = {{\mbox{}\\\url{http://wstein.org/papers/generating_hecke/}}} } or \bibitem[AS02]{generatinghecke} A.~Agashe and W.\thinspace{}A. Stein, \emph{Generating the \protect{H}ecke algebra (appendix to {L}ario-{S}choof)}, Experiment. Math. \textbf{11} (2002), no.~2, 303--311, {\mbox{}\\\url{http://wstein.org/papers/generating_hecke/}}.