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Algorithms for Chow-Heegner points via iterated integrals


Authors: Henri Darmon, Michael Daub, Sam Lichtenstein and Victor Rotger
Journal: Math. Comp. 84 (2015), 2505-2547
MSC (2010): Primary 11F67, 11G05, 11Y16, 14C15
DOI: https://doi.org/10.1090/S0025-5718-2015-02927-5
Published electronically: March 2, 2015
MathSciNet review: 3356037
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Abstract: Let $ E_{/\mathbf Q}$ be an elliptic curve of conductor $ N$ and let $ f$ be the weight $ 2$ newform on $ \Gamma _0(N)$ associated to it by modularity. Building on an idea of S. Zhang, an article by Darmon, Rotger, and Sols describes the construction of so-called Chow-Heegner points, $ P_{T,f}\in E({\bar {\mathbf Q}})$, indexed by algebraic correspondences $ T\subset X_0(N)\times X_0(N)$. It also gives an analytic formula, depending only on the image of $ T$ in cohomology under the complex cycle class map, for calculating $ P_{T,f}$ numerically via Chen's theory of iterated integrals. The present work describes an algorithm based on this formula for computing the Chow-Heegner points to arbitrarily high complex accuracy, carries out the computation for all elliptic curves of rank $ 1$ and conductor $ N< 100$ when the cycles $ T$ arise from Hecke correspondences, and discusses several important variants of the basic construction.


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  • [ARS06] Amod Agashe, Kenneth Ribet, and William A. Stein, The Manin constant, Pure Appl. Math. Q. 2 (2006), no. 2, 617-636. MR 2251484 (2007c:11076), https://doi.org/10.4310/PAMQ.2006.v2.n2.a11
  • [BD1] Massimo Bertolini and Henri Darmon, Kolyvagin's descent and Mordell-Weil groups over ring class fields, J. Reine Angew. Math. 412 (1990), 63-74. MR 1079001 (91j:11048), https://doi.org/10.1515/crll.1990.412.63
  • [BD2] Massimo Bertolini and Henri Darmon, Heegner points, $ p$-adic $ L$-functions, and the Cerednik-Drinfeld uniformization, Invent. Math. 131 (1998), no. 3, 453-491. MR 1614543 (99f:11080), https://doi.org/10.1007/s002220050211
  • [BDP1] Massimo Bertolini, Henri Darmon, and Kartik Prasanna, Generalized Heegner cycles and $ p$-adic Rankin $ L$-series, Duke Math. J. 162 (2013), no. 6, 1033-1148. With an appendix by Brian Conrad. MR 3053566, https://doi.org/10.1215/00127094-2142056
  • [BDP2] Massimo Bertolini, Henri Darmon, and Kartik Prasanna, Chow-Heegner points on CM elliptic curves and values of $ p$-adic $ L$-series, Int. Math. Res. Not. IMRN 3 (2014), 745-793. MR 3163566
  • [Bi] Bryan Birch, Heegner points: the beginnings, Heegner points and Rankin $ L$-series, Math. Sci. Res. Inst. Publ., vol. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 1-10. MR 2083207 (2005d:11083), https://doi.org/10.1017/CBO9780511756375.002
  • [BG] Bryan Birch and Benedict Gross, Correspondence, Heegner points and Rankin $ L$-series, Math. Sci. Res. Inst. Publ., vol. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 11-23. MR 2083208 (2005g:11120), https://doi.org/10.1017/CBO9780511756375.003
  • [BO] K. Bringmann, K. Ono, Coefficients of harmonic weak Maass forms, Proceedings of the 2008 University of Florida conference on partitions, $ q$-series, and modular forms.
  • [BrPh] Nicolas Brisebarre and Georges Philibert, Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant $ j$, J. Ramanujan Math. Soc. 20 (2005), no. 4, 255-282. MR 2193216 (2006k:11074)
  • [Ch] Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831-879. MR 0454968 (56 #13210)
  • [Cr] J. Cremona, Chapter 2: Modular Symbol Algorithms, in Algorithms for Modular Elliptic curves. http://www.warwick.ac.uk/~masgaj/book/fulltext/index.html.
  • [Cre] J.E. Cremona, Elliptic Curves Data, http://www.warwick.ac.uk/~masgaj/ftp/data/.
  • [CWZ] J. A. Csirik, J. L. Wetherell, M. E. Zieve, On the genera of $ X_0(N)$, preprint arXiv:math/0006096.
  • [D1] Henri Darmon, Integration on $ \mathcal {H}_p\times \mathcal {H}$ and arithmetic applications, Ann. of Math. (2) 154 (2001), no. 3, 589-639. MR 1884617 (2003j:11067), https://doi.org/10.2307/3062142
  • [D2] Henri Darmon, Rational Points on Modular Elliptic Curves, CBMS Regional Conference Series in Mathematics, vol. 101, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004. MR 2020572 (2004k:11103)
  • [D3] H. Darmon, Cycles on modular varieties and rational points on elliptic curves, in Explicit Methods in Number Theory, Oberwolfach reports 6:3 (2009), 1843-1920.
  • [DL] Henri Darmon and Adam Logan, Periods of Hilbert modular forms and rational points on elliptic curves, Int. Math. Res. Not. 40 (2003), 2153-2180. MR 1997296 (2005f:11110), https://doi.org/10.1155/S1073792803131108
  • [DP] Henri Darmon and Robert Pollack, Efficient calculation of Stark-Heegner points via overconvergent modular symbols, Israel J. Math. 153 (2006), 319-354. MR 2254648 (2007k:11077), https://doi.org/10.1007/BF02771789
  • [DLR] H. Darmon, A. Lauder, V. Rotger, Stark points and $ p$-adic iterated integrals attached to modular forms of weight one, submitted.
  • [DRS] Henri Darmon, Victor Rotger, and Ignacio Sols, Iterated integrals, diagonal cycles and rational points on elliptic curves, Publications mathématiques de Besançon. Algèbre et théorie des nombres, 2012/2, Publ. Math. Besançon Algèbre Théorie Nr., vol. 2012/, Presses Univ. Franche-Comté, Besançon, 2012, pp. 19-46 (English, with English and French summaries). MR 3074917
  • [DR] H. Darmon, V. Rotger, Diagonal cycles and Euler systems I: A $ p$-adic Gross-Zagier formula, Annales Sc. École Normal Supérieure, 47 (2014), no. 4, 779-832. MR 3250064
  • [Dau13] M. Daub, Complex and $ p$-adic computations of Chow-Heegner points, Ph.D. Thesis, University of California, Berkeley.
  • [Del02] C. Delaunay, Formes modulaires et invariants de courbes elliptiques définies sur $ \mathbf {Q}$, Thèse de Doctorat, Université Bordeaux I, available at http://math.univ-lyon1.fr/~delaunay/.
  • [DS] Fred Diamond and Jerry Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196 (2006f:11045)
  • [Dok04] Tim Dokchitser, Computing special values of motivic $ L$-functions, Experiment. Math. 13 (2004), no. 2, 137-149. MR 2068888 (2005f:11128)
  • [E] Martin Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. 5 (1954), 355-366 (German). MR 0063406 (16,116d)
  • [G] Josep Gonzàlez Rovira, Equations of hyperelliptic modular curves, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 779-795 (English, with French summary). MR 1150566 (93g:11064)
  • [Gr] Matthew Greenberg, Heegner Points and Rigid Analytic Modular Forms, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)-McGill University (Canada). MR 2710023
  • [GJP$^+$09] Grigor Grigorov, Andrei Jorza, Stefan Patrikis, William A. Stein, and Corina Tarniţa, Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves, Math. Comp. 78 (2009), no. 268, 2397-2425. MR 2521294 (2010g:11106), https://doi.org/10.1090/S0025-5718-09-02253-4
  • [Gro91] Benedict H. Gross, Kolyvagin's work on modular elliptic curves, $ L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235-256. MR 1110395 (93c:11039), https://doi.org/10.1017/CBO9780511526053.009
  • [GK92] Benedict H. Gross and Stephen S. Kudla, Heights and the central critical values of triple product $ L$-functions, Compositio Math. 81 (1992), no. 2, 143-209. MR 1145805 (93g:11047)
  • [GrZa] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $ L$-series, Invent. Math. 84 (1986), no. 2, 225-320. MR 833192 (87j:11057), https://doi.org/10.1007/BF01388809
  • [H1] R. Hain, Lectures on the Hodge-de Rham theory of the fundamental group of $ \mathbf P^1-\{0,1,\infty \}$. http://math.arizona.edu/~swc/notes/files/05HainNotes.pdf.
  • [H2] Richard M. Hain, The geometry of the mixed Hodge structure on the fundamental group, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 247-282. MR 927984 (89g:14010)
  • [Ka] Ernst Kani, Endomorphisms of Jacobians of modular curves, Arch. Math. (Basel) 91 (2008), no. 3, 226-237. MR 2439596 (2009j:11097), https://doi.org/10.1007/s00013-008-2696-7
  • [Ko] V. A. Kolyvagin, Finiteness of $ E({\bf Q})$ and SH $ (E,{\bf Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522-540, 670-671 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 523-541. MR 954295 (89m:11056)
  • [La] A. Lauder, Efficient computation of Rankin $ p$-adic $ L$-functions, in Computations with Modular Forms, Proceedings of a Summer School and Conference, Heidelberg, August/September 2011, Boeckle G. and Wiese G. (eds), Springer Verlag, to appear.
  • [LoWe] David Loeffler and Jared Weinstein, On the computation of local components of a newform, Math. Comp. 81 (2012), no. 278, 1179-1200. MR 2869056 (2012k:11064), https://doi.org/10.1090/S0025-5718-2011-02530-5
  • [Mal] Gregorio Malajovich, Condition number bounds for problems with integer coefficients, J. Complexity 16 (2000), no. 3, 529-551. Complexity theory, real machines, and homotopy (Oxford, 1999). MR 1787884 (2002g:65180), https://doi.org/10.1006/jcom.2000.0552
  • [Man] Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19-66 (Russian). MR 0314846 (47 #3396)
  • [MSD74] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1-61. MR 0354674 (50 #7152)
  • [O] A. P. Ogg, On the Weierstrass points of $ X_{0}(N)$, Illinois J. Math. 22 (1978), no. 1, 31-35. MR 0463178 (57 #3136)
  • [Pr90] Dipendra Prasad, Trilinear forms for representations of $ {\rm GL}(2)$ and local $ \epsilon $-factors, Compositio Math. 75 (1990), no. 1, 1-46. MR 1059954 (91i:22023)
  • [R] Hans Rademacher, The Fourier coefficients of the modular invariant J($ \tau $), Amer. J. Math. 60 (1938), no. 2, 501-512. MR 1507331, https://doi.org/10.2307/2371313
  • [RS] K. A. Ribet, W. A. Stein, Lectures on modular forms and Hecke operators, available at http://wstein.org/books.
  • [Sil] Joseph H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)
  • [S$^+$09] W.A. Stein et al., Sage Mathematics Software (Version 4.7.1), The Sage Development Team, 2011, http://www.sagemath.org.
  • [Stn] William Stein, Modular Forms, a Computational Approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells. MR 2289048 (2008d:11037)
  • [Stv] Glenn Stevens, Arithmetic on Modular Curves, Progress in Mathematics, vol. 20, Birkhäuser Boston Inc., Boston, MA, 1982. MR 670070 (87b:11050)
  • [Wa] Michel Waldschmidt, Nombres Transcendants et Groupes Algébriques, Astérisque, vol. 69, Société Mathématique de France, Paris, 1979 (French). With appendices by Daniel Bertrand and Jean-Pierre Serre; With an English summary. MR 570648 (82k:10041)
  • [Wat02] Mark Watkins, Computing the modular degree of an elliptic curve, Experiment. Math. 11 (2002), no. 4, 487-502 (2003). MR 1969641 (2004c:11091)
  • [YZZ] X. Yuan, S. Zhang, W. Zhang, Triple product L-series and Gross-Schoen cycles I: split case, preprint.
  • [Zh] Shouwu Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), no. 1, 27-147. MR 1826411 (2002g:11081), https://doi.org/10.2307/2661372

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Additional Information

Henri Darmon
Affiliation: Department of Mathematics, McGill University, Montreal, Canada
Email: darmon@math.mcgill.ca

Michael Daub
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California
Email: mwdaub@math.berkeley.edu

Sam Lichtenstein
Affiliation: University of California at Stanford, Stanford, California
Email: saml@math.stanford.edu

Victor Rotger
Affiliation: Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
Email: victor.rotger@upc.edu

DOI: https://doi.org/10.1090/S0025-5718-2015-02927-5
Received by editor(s): December 23, 2011
Received by editor(s) in revised form: October 28, 2013, and December 23, 2013
Published electronically: March 2, 2015
Dedicated: With an appendix by William Stein
Article copyright: © Copyright 2015 American Mathematical Society

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