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

  • Matija KazalickiEmail author


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.


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


Author's contributions


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., and by the Croatian Science Foundation under the Project No. IP-2018-01-1313.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

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