Congruences for sporadic sequences, three fold covers of the elliptic modular surfaces and modular forms for non-congruence subgroups

Abstract: In 1979, in the course of the proof of the irrationality of $\zeta(2)$ Ap\'ery introduced numbers $b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}$ that are, surprisingly, integral solutions of recursive relations $$(n+1)^2 u_{n+1} - (11n^2+11n+3)u_n-n^2u_{n-1} = 0.$$ Zagier performed a computer search on first 100 million triples $(A,B,C)\in \mathbb{Z}^3$ and found that the recursive relation generalizing $b_n$ $$(n+1)u_{n+1} - (An^2+An+B)u_n + C n ^2 u_{n-1}=0,$$ 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 the for prime $p\ge 5$ \begin{equation*} b_{(p-1)/2} \equiv \begin{cases} 4a^2-2p \pmod{p} \textrm{ if } p = a^2+b^2,\textrm{ a odd}\\ 0 \pmod{p} \textrm{ if } p\equiv 3 \pmod{4}.\end{cases} \end{equation*}

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 we proved the congruence for the sixth sequence.

In this talk we describe congruences for the Ap\'ery numbers $b_{(p-1)/3}$ (and also for the other sporadic sequences). For that we study Atkin and Swinnerton-Dyer type of congruences between Fourier coefficients of cusp forms for non-congruence subgroups, $L$-functions of three covers of elliptic modular surfaces and Galois representations attached to these covers.

number theory

Audience: researchers in the topic

( video )

Warsaw Number Theory Seminar