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Congruences for sporadic sequences and modular forms for non-congruence subgroups

  • Matija KazalickiEmail author
Research
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Abstract

In 1979, in the course of the proof of the irrationality of \(\zeta (2)\) Apéry introduced numbers \(b_n\) that are, surprisingly, integral solutions of the recursive relation
$$\begin{aligned} (n+1)^2 u_{n+1} - (11n^2+11n+3)u_n-n^2u_{n-1} = 0. \end{aligned}$$
Indeed, \(b_n\) can be expressed as \(b_n= \sum _{k=0}^n {n \atopwithdelims ()k}^2{n+k \atopwithdelims ()k}\). Zagier performed a computer search of the first 100 million triples \((A,B,C)\in {\mathbb {Z}}^3\) and found that the recursive relation generalizing \(b_n\)
$$\begin{aligned} (n+1)^2 u_{n+1} - (An^2+An+B)u_n + C n ^2 u_{n-1}=0, \end{aligned}$$
with the initial conditions \(u_{-1}=0\) and \(u_0=1\) has (non-degenerate, i.e., \(C(A^2-4C)\ne 0\)) integral solution for only six more triples (whose solutions are so-called sporadic sequences). Stienstra and Beukers showed that for the prime \(p\ge 5\)
$$\begin{aligned} b_{(p-1)/2} \equiv {\left\{ \begin{array}{ll} 4a^2-2p \pmod {p} \text { if } p = a^2+b^2, \text { a odd}\\ 0 \pmod {p} \text { if } p\equiv 3 \pmod {4}.\end{array}\right. } \end{aligned}$$
Recently, Osburn and Straub proved similar congruences for all but one of the six Zagier’s sporadic sequences (three cases were already known to be true by the work of Stienstra and Beukers) and conjectured the congruence for the sixth sequence [which is a solution of the recursion determined by triple (17, 6, 72)]. In this paper, we prove that the remaining congruence by studying Atkin and Swinnerton-Dyer congruences between Fourier coefficients of a certain cusp forms for non-congruence subgroup.

Keywords

Apéry numbers Modular forms Atkin and Swinnerton-Dyer congruences Non-congruence subgroups 

Notes

Author's contributions

Acknowledgements

The author would like to thank Robert Osburn and Armin Straub for bringing this problem to his attention. Also, the author would like to thank the referees for their valuable comments which helped to improve the paper. The author was supported by the QuantiXLie Centre of Excellence, a Project co-financed by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004), and by the Croatian Science Foundation under the Project No. IP-2018-01-1313.

References

  1. 1.
    Apéry, R.: Irrationalité de \(\zeta (2)\) et \(\zeta (3)\), Astérisque. In: Luminy Conference on Arithmetic, pp. 11–13 (1979)Google Scholar
  2. 2.
    Atkin, A.O.L., Swinnerton-Dyer, H.P.F.: Modular forms on noncongruence subgroups. In: Combinatorics (Proceedings of Symposia in Pure Mathematics, vol. XIX, pp. 1–25. University of California, Los Angeles, 1968). American Mathematical Society, Providence (1971)Google Scholar
  3. 3.
    Beukers, F.: Another congruence for the Apéry numbers. J. Number Theory 25, 201–210 (1987)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bosma, W., Cannon, J., Playoust, C.: The magma algebra system. I. The user language. Computational algebra and number theory (London, 1993). J. Symb. Comput. 24, 235–265 (1997)CrossRefGoogle Scholar
  5. 5.
    Deligne, P.: Formes modulaires et représentations \(l\)-adiques. In: Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363. Lecture Notes in Mathematics, vol. 175, pp. 139–172. Springer, Berlin (1971)Google Scholar
  6. 6.
    Dujella, A., Kazalicki, M.: Diophantine m-tuples in finite fields and modular forms. arXiv e-prints arXiv:1609.09356 (2016)
  7. 7.
    Kazalicki, M.: Modular forms, hypergeometric functions and congruences. Ramanujan J. 34, 1–9 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kazalicki, M., Scholl, A.J.: Modular forms, de Rham cohomology and congruences. Trans. Am. Math. Soc. 368, 7097–7117 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, W.-C.W., Long, L., Yang, Z.: Modular forms for noncongruence subgroups. Q. J. Pure Appl. Math. 1, 205–221 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, W.-C.W., Long, L., Yang, Z.: On Atkin–Swinnerton-Dyer congruence relations. J. Number Theory 113, 117–148 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Osburn, R., Straub, A.: Interpolated sequences and critical L-values of modular forms. In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Texts and Monographs in Symbolic Computation, pp. 327–349. Springer, Vienna (2019)Google Scholar
  12. 12.
    Scholl, A.J.: Modular forms and de Rham cohomology; Atkin–Swinnerton-Dyer congruences. Invent. Math. 79, 49–77 (1985)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Scholl, A.J.: The \(l\)-adic representations attached to a certain noncongruence subgroup. J. Reine Angew. Math. 392, 1–15 (1988)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Shimura, G.: Introduction to the arithmetic theory of automorphic functions. In: Kanô Memorial Lectures, vol. 1. Publications of the Mathematical Society of Japan, Iwanami Shoten, Publishers, Tokyo, Princeton University Press, Princeton (1971)Google Scholar
  15. 15.
    Stienstra, J., Beukers, F.: On the Picard–Fuchs equation and the formal Brauer group of certain elliptic \(K3\)-surfaces. Math. Ann. 271, 269–304 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    The LMFDB Collaboration: The \(L\)-Functions and Modular Forms Database. http://www.lmfdb.org (2018). Accessed 15 Dec 2018
  17. 17.
    The Sage Developers: SageMath, the Sage Mathematics Software System (Version 7.2). http://www.sagemath.org (2018). Accessed 15 Dec 2018
  18. 18.
    Verrill, H.A.: Congruences related to modular forms. Int. J. Number Theory 6, 1367–1390 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zagier, D.: Integral solutions of Apéry-like recurrence equations. In: Groups and Symmetries. CRM Proceedings Lecture Notes, vol. 47, pp. 349–366. American Mathematical Society, Providence (2009)Google Scholar
  20. 20.
    Zagier, D.: Arithmetic and topology of differential equations. In: Proceedings of the 2016 ECM (2017)Google Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

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