Evaluating modular forms on Shimura curves
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Abstract:
Let $f$ be a newform, as specified by its Hecke eigenvalues, on a Shimura curve $X$. We describe a method for evaluating $f$. The most interesting case is when $X$ arises as a compact quotient of the hyperbolic plane, so that classical $q$-expansions are not available. The method takes the form of an explicit, rapidly-convergent formula that is well-suited for numerical computation. We apply it to the problem of computing modular parametrizations of elliptic curves, and illustrate with some numerical examples.
An important ingredient is a new method for numerically computing Petersson inner products, which may be of independent interest.
References
- A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701
- Andrew R. Booker, Andreas Strömbergsson, and Akshay Venkatesh, Effective computation of Maass cusp forms, Int. Math. Res. Not. , posted on (2006), Art. ID 71281, 34. MR 2249995, DOI 10.1155/IMRN/2006/71281
- Henri Carayol, Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), no. 2, 151–230 (French). MR 860139
- Henri Darmon and Robert Pollack, Efficient calculation of Stark-Heegner points via overconvergent modular symbols, Israel J. Math. 153 (2006), 319–354. MR 2254648, DOI 10.1007/BF02771789
- Henri Darmon and Victor Rotger, Algebraic cycles and Stark-Heegner points, http://www-ma2.upc.edu/vrotger/docs/AWS2011/aws.pdf, 2011.
- Lassina Dembélé, Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms, Math. Comp. 76 (2007), no. 258, 1039–1057. MR 2291849, DOI 10.1090/S0025-5718-06-01914-4
- Lassina Dembélé and Steve Donnelly, Computing Hilbert modular forms over fields with nontrivial class group, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 371–386. MR 2467859, DOI 10.1007/978-3-540-79456-1_{2}5
- Noam D. Elkies, Heegner point computations, Algorithmic number theory (Ithaca, NY, 1994) Lecture Notes in Comput. Sci., vol. 877, Springer, Berlin, 1994, pp. 122–133. MR 1322717, DOI 10.1007/3-540-58691-1_{4}9
- Noam D. Elkies, Shimura curve computations, Algorithmic number theory (Portland, OR, 1998) Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, pp. 1–47. MR 1726059, DOI 10.1007/BFb0054850
- Dorian Goldfeld, The Gauss class number problem for imaginary quadratic fields, Heegner points and Rankin $L$-series, Math. Sci. Res. Inst. Publ., vol. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 25–36. MR 2083209, DOI 10.1017/CBO9780511756375.004
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
- Matthew Greenberg, Heegner point computations via numerical $p$-adic integration, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 361–376. MR 2282936, DOI 10.1007/11792086_{2}6
- Matthew Greenberg, Heegner points and rigid analytic modular forms, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–McGill University (Canada). MR 2710023
- Matthew Greenberg and John Voight, Computing systems of Hecke eigenvalues associated to Hilbert modular forms, Math. Comp. 80 (2011), no. 274, 1071–1092. MR 2772112, DOI 10.1090/S0025-5718-2010-02423-8
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
- Benedict H. Gross, Heegner points on $X_0(N)$, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 87–105. MR 803364
- 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, DOI 10.1017/CBO9780511526053.009
- Paul E. Gunnells and Dan Yasaki, Hecke operators and Hilbert modular forms, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 387–401. MR 2467860, DOI 10.1007/978-3-540-79456-1_{2}6
- Michael Harris and Stephen S. Kudla, The central critical value of a triple product $L$-function, Ann. of Math. (2) 133 (1991), no. 3, 605–672. MR 1109355, DOI 10.2307/2944321
- Roman Holowinsky, Sieving for mass equidistribution, Ann. of Math. (2) 172 (2010), no. 2, 1499–1516. MR 2680498, DOI 10.4007/annals.2010.172.1499
- Henryk Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. MR 1942691, DOI 10.1090/gsm/053
- Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
- Dimitar Jetchev, Kristin Lauter, and William Stein, Explicit Heegner points: Kolyvagin’s conjecture and non-trivial elements in the Shafarevich-Tate group, J. Number Theory 129 (2009), no. 2, 284–302. MR 2473878, DOI 10.1016/j.jnt.2008.05.007
- David R. Kohel and Helena A. Verrill, Fundamental domains for Shimura curves, J. Théor. Nombres Bordeaux 15 (2003), no. 1, 205–222 (English, with English and French summaries). Les XXIIèmes Journées Arithmetiques (Lille, 2001). MR 2019012
- P. D. Nelson, A. Pitale, and A. Saha, Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels, ArXiv e-prints, May 2012.
- P. D. Nelson, Computing on Shimura curves II: Implementing the Shimizu lifting, Appendix C of the notes Periods and special values of $L$-functions, http://swc.math.arizona.edu/aws/2011/2011PrasannaNotesProject.pdf, 2011.
- Arnold Pizer, An algorithm for computing modular forms on $\Gamma _{0}(N)$, J. Algebra 64 (1980), no. 2, 340–390. MR 579066, DOI 10.1016/0021-8693(80)90151-9
- K. Prasanna, Computing on Shimura curves I: Expansions at CM points, Appendix B of the notes Periods and special values of $L$-functions, http://swc.math.arizona.edu/aws/2011/2011PrasannaNotesProject.pdf, 2011.
- Hideo Shimizu, On zeta functions of quaternion algebras, Ann. of Math. (2) 81 (1965), 166–193. MR 171771, DOI 10.2307/1970389
- Hideo Shimizu, Theta series and automorphic forms on $\textrm {GL}_{2}$, J. Math. Soc. Japan 24 (1972), 638–683. MR 333081, DOI 10.2969/jmsj/02440638
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
- 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, DOI 10.1090/gsm/079
- W. A. Stein et al., Sage Mathematics Software (Version 4.1.1), The Sage Development Team, 2009. http://www.sagemath.org.
- Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
- J. Voight and J. Willis, Computing power series expansions of modular forms, ArXiv e-prints, April 2012.
- John Voight, Computing CM points on Shimura curves arising from cocompact arithmetic triangle groups, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 406–420. MR 2282939, DOI 10.1007/11792086_{2}9
- John Voight, Computing fundamental domains for Fuchsian groups, J. Théor. Nombres Bordeaux 21 (2009), no. 2, 469–491 (English, with English and French summaries). MR 2541438
- John Voight, Computing automorphic forms on Shimura curves over fields with arbitrary class number, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 357–371. MR 2721432, DOI 10.1007/978-3-642-14518-6_{2}8
- Thomas Crawford Watson, Rankin triple products and quantum chaos, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Princeton University. MR 2703041
- Shouwu Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), no. 1, 27–147. MR 1826411, DOI 10.2307/2661372
Additional Information
- Paul D. Nelson
- Affiliation: EPFL, Station 8, CH-1015 Lausanne, Switzerland
- Address at time of publication: ETH Zurich, Raemistrasse 101, 8092 Zurich, Switzerland
- Email: Paul.nelson@math.ethz.ch
- Received by editor(s): October 9, 2012
- Received by editor(s) in revised form: October 10, 2013
- Published electronically: January 23, 2015
- Additional Notes: The author was supported by NSF grant OISE-1064866 and partially supported by grant SNF-137488 during the completion of this paper.
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2471-2503
- MSC (2010): Primary 11F11, 11Y40; Secondary 11F27
- DOI: https://doi.org/10.1090/S0025-5718-2015-02943-3
- MathSciNet review: 3356036