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

2019/01/10 by Matija Kazalicki, Kazalicki, Matija
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1901.03098

openalex publication_date 2019/01/10 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

In the course of the proof of the irrationality of zeta(2) R. Apery\nintroduced numbers b_n = \∑_k=0ⁿ n choose k²n+k choose k. Stienstra\nand Beukers showed that for the prime p > 3 Apery numbers satisfy congruence\nb((p-1)/2) = 4a²-2p mod p, if p = a²+b² (where a is odd). Later, Zagier\nfound some generalizations of Apery numbers, so called sporadic sequences, and\nrecently Osburn and Straub proved similar congruences for all but one of the\nsix Zagier's sporadic sequences (three cases were already known to be true) and\nconjectured the congruence for the sixth sequence.\n In this paper we prove that remaining congruence by studying Atkin and\nSwinnerton-Dyer congruences between Fourier coefficients of certain cusp form\nfor non-congurence subgroup.\n

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