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

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.