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

## TL;DR

This paper proves a conjectured congruence relation for a sporadic sequence related to Apery numbers by analyzing Atkin and Swinnerton-Dyer congruences in the context of non-congruence modular forms.

## Contribution

It establishes the remaining congruence for Zagier's sporadic sequences using Fourier coefficients of non-congruence cusp forms, completing previous partial results.

## Key findings

- Proved the conjectured congruence for the sixth sporadic sequence.
- Demonstrated Atkin and Swinnerton-Dyer congruences for non-congruence modular forms.
- Connected sporadic sequences with non-congruence modular forms through congruence relations.

## Abstract

In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) = 4a^2-2p mod p, if p = a^2+b^2 (where a is odd). Later, Zagier found some generalizations of Apery numbers, so called sporadic sequences, and 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) and conjectured the congruence for the sixth sequence. In this paper we prove that remaining congruence by studying Atkin and Swinnerton-Dyer congruences between Fourier coefficients of certain cusp form for non-congurence subgroup.