Skip to main content
SpringerLink
Log in

Computing Classical Modular Forms for Arbitrary Congruence Subgroups

  • Conference paper
  • First Online:
Arithmetic Geometry, Number Theory, and Computation

Part of the book series: Simons Symposia ((SISY))

Abstract

In this paper, we prove the existence of an efficient algorithm for the computation of systems of Hecke eigenvalues of modular forms of weight k and level Γ, where \(\Gamma \subseteq SL_{2}({\mathbb {Z}})\) is an arbitrary congruence subgroup. We also discuss some practical aspects and provide the necessary theoretical background.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Eran Assaf. Dimensions of the decomposition of the Jacobians of \({X}_{ns}^{+}(p)\) for primes p < 100 and characteristic polynomials of the Hecke operators T p. https://github.com/assaferan/ModularSymbols/blob/master/dec_dims_char_polys_ns_cartan.dat. Last accessed: 2020-02-03.

  2. Eran Assaf. Modular symbols package for arbitrary congruence subgroups. https://github.com/assaferan/ModularSymbols/. Last accessed: 2020-02-03.

  3. A Oliver L Atkin et al. Twists of newforms and pseudo-eigenvalues of W-operators. Inventiones mathematicae, 48(3):221–243, 1978.

    Article  MathSciNet  Google Scholar 

  4. Barinder Banwait and John Cremona. Tetrahedral elliptic curves and the local-global principle for isogenies. Algebra & Number Theory, 8(5):1201–1229, 2014.

    Article  MathSciNet  Google Scholar 

  5. Burcu Baran. Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem. Journal of Number Theory, 130(12):2753–2772, 2010.

    Article  MathSciNet  Google Scholar 

  6. Burcu Baran. An exceptional isomorphism between modular curves of level 13. Journal of Number Theory, 145:273–300, 2014.

    Article  MathSciNet  Google Scholar 

  7. Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John E. Cremona, Maarten Derickx, Min Lee, David Lowry-Duda, David Roe, Andrew V. Sutherland, and John Voight. Computing classical modular forms. In Arithmetic Geometry, Number Theory, and Computation, pages 131--214. Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-80914-0_4

  8. Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system I: The user language. Journal of Symbolic Computation, 24(3–4):235–265, 1997.

    Article  MathSciNet  Google Scholar 

  9. Josha Box. Computing models for quotients of modular curves. Res. Number Theory 7(3):51, 2021.

    Google Scholar 

  10. Kevin Buzzard. Computing weight one modular forms over \(\mathbb {C}\) and \(\overline {\mathbb {f}}_{p}\). In Computations with modular forms, pages 129–146. Springer, 2014.

    Google Scholar 

  11. Jean-Marc Couveignes and Bas Edixhoven. Computational aspects of modular forms and Galois representations. Princeton University Press, 2011.

    MATH  Google Scholar 

  12. John E Cremona. Algorithms for modular elliptic curves. 2nd edition. Cambridge University Press, Cambridge, 1997.

    Google Scholar 

  13. Chris Cummins and Sebastian Pauli. Congruence subgroups of \({P}{S}{L}_2(\mathbb {Z})\). https://mathstats.uncg.edu/sites/pauli/congruence/csg6.html. Last accessed: 2020-02-03.

  14. Fred Diamond and Jerry Michael Shurman. A first course in modular forms, volume 228. Springer, 2005.

    Google Scholar 

  15. Valerio Dose, Julio Fernández, Josep González, and René Schoof. The automorphism group of the non-split Cartan modular curve of level 11. Journal of Algebra, 417:95–102, 2014.

    Article  MathSciNet  Google Scholar 

  16. Enrique González-Jiménez and Roger Oyono. Non-hyperelliptic modular curves of genus 3. Journal of Number Theory, 130(4):862–878, 2010.

    Article  MathSciNet  Google Scholar 

  17. Lloyd JP Kilford. Modular forms: a classical and computational introduction. Mathematical intelligencer, 32(3):58–59, 2010.

    Google Scholar 

  18. Ravi S Kulkarni. An arithmetic-geometric method in the study of the subgroups of the modular group. American Journal of mathematics, 113(6):1053–1133, 1991.

    Google Scholar 

  19. Ju I Manin. Parabolic points and zeta-functions of modular curves. Mathematics of the USSR-Izvestiya, 6(1):19, 1972.

    Google Scholar 

  20. Barry Mazur. Rational points on modular curves. In Modular functions of one variable V, pages 107–148. Springer, 1977.

    Google Scholar 

  21. Pietro Mercuri and René Schoof. Modular forms invariant under non-split Cartan subgroups. Mathematics of Computation, 89(324):1969–1991, 2020.

    Article  MathSciNet  Google Scholar 

  22. Loïc Merel. Opérateurs de Hecke pour Γ0(N) et fractions continues. In Annales de l’institut Fourier, volume 41, pages 519–537, 1991.

    Google Scholar 

  23. Loïc Merel. Universal Fourier expansions of modular forms. In On Artin’s Conjecture for Odd 2-Dimensional Representations, pages 59–94. Springer, 1994.

    Google Scholar 

  24. James S Milne. Introduction to Shimura varieties. Harmonic analysis, the trace formula, and Shimura varieties, 4:265–378, 2005.

    Google Scholar 

  25. Andrew Ogg. Modular forms and Dirichlet series. WA Benjamin New York, 1969.

    Google Scholar 

  26. Jeremy Rouse and David Zureick-Brown. Elliptic curves over \(\mathbb {Q} \) and 2-adic images of galois. Research in Number Theory, 1(1):12, 2015.

    Google Scholar 

  27. George J Schaeffer. Hecke stability and weight 1 modular forms. Mathematische Zeitschrift, 281(1-2):159–191, 2015.

    Google Scholar 

  28. Jean-Pierre Serre. Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. math, 15:259–331, 1972.

    Article  MathSciNet  Google Scholar 

  29. Goro Shimura. Introduction to the arithmetic theory of automorphic functions, volume 1. Princeton University Press, 1971.

    MATH  Google Scholar 

  30. Vyacheslav Vladimirovich Shokurov. The study of the homology of Kuga varieties. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 44(2):443–464, 1980.

    Google Scholar 

  31. William A Stein. An introduction to computing modular forms using modular symbols. In An MSRI Proceedings, 2002.

    Google Scholar 

  32. William A Stein. Modular forms, a computational approach, volume 79. American Mathematical Soc., 2007.

    Google Scholar 

  33. William Arthur Stein. Explicit approaches to modular abelian varieties. PhD thesis, University of California, Berkeley, 2000.

    Google Scholar 

  34. Andrew Sutherland and David Zywina. Modular curves of prime-power level with infinitely many rational points. Algebra & Number Theory, 11(5):1199–1229, 2017.

    Article  MathSciNet  Google Scholar 

  35. John A Todd and Harold SM Coxeter. A practical method for enumerating cosets of a finite abstract group. Proceedings of the Edinburgh Mathematical Society, 5(1):26–34, 1936.

    Google Scholar 

  36. John Voight. Quaternion Algebras. Graduate Texts in Mathematics, 288. Springer, 2021.

    Google Scholar 

  37. Gabor Wiese et al. Modular forms of weight one over finite fields. PhD thesis, Stieltjes Institute for Mathematics, Faculty of Mathematics and Natural Sciences, 2005.

    Google Scholar 

  38. David Zywina. Computing actions on cusp forms. arXiv preprint arXiv:2001.07270, 2020.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dartmouth College, Hanover, NH, USA

    Eran Assaf

Authors
  1. Eran Assaf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eran Assaf .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Jennifer S. Balakrishnan

  2. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Noam Elkies

  3. Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI, USA

    Brendan Hassett

  4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Bjorn Poonen

  5. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrew V. Sutherland

  6. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    John Voight

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Assaf, E. (2021). Computing Classical Modular Forms for Arbitrary Congruence Subgroups. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-80914-0_2

Download citation

Publish with us

Policies and ethics

Search

Navigation