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.

中文翻译：

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零散序列的同余和非同余子群的模块化形式

1979年，在证明\（\ zeta（2）\）的不合理性的过程中，Apéry引入了数字\（b_n \），这令人惊讶地是递归关系$$ \ begin {aligned}（n +1）^ 2 u_ {n + 1}-（11n ^ 2 + 11n + 3）u_n-n ^ 2u_ {n-1} =0。\ end {aligned} $$确实，\（b_n \）可以是表示为\（b_n = \ sum _ {k = 0} ^ n {n \ atopwithdelims（）k} ^ 2 {n + k \ atopwithdelims（）k} \）。Zagier对{\ mathbb {Z}} ^ 3 \中的前1亿个三元组\（（A，B，C）\）进行了计算机搜索，发现递归关系推广了\（b_n \）$$ \ begin {已对齐}（n + 1）^ 2 u_ {n + 1}-（An ^ 2 + An + B）u_n + C n ^ 2 u_ {n-1} = 0，\ end {aligned} $$与初始条件\（u _ {-1} = 0 \）和\（u_0 = 1 \）仅具有六个三元组（不退化，即\（C（A ^ 2-4C）\ ne 0 \））积分解（其解决方案是所谓的零星序列）。Stienstra和Beukers证明对于素数\（p \ ge 5 \）$$ \ begin {aligned} b _ {（p-1）/ 2} \ equiv {\ left \ {\ begin {array} {ll} 4a ^ 2-2p \ pmod {p} \ text {如果} p = a ^ 2 + b ^ 2，\ text {奇数} \\ 0 \ pmod {p} \ text {如果} p \ equiv 3 \ pmod {4 }。\ end {array} \ right。} \ end {aligned} $$最近，Osburn和Straub在六个Zagier散发序列中都证明了相似的同余（Stienstra和Beukers的工作已经知道三个案例是正确的），并猜想第六个序列的同余[这是对递归由三元组（17，6，72）确定]。在本文中，我们通过研究非一致性子群的某个尖点形式的傅立叶系数之间的Atkin和Swinnerton-Dyer一致性，证明了剩余的一致性。