On Atkin and Swinnerton-Dyer Congruence Relations (2)

Atkin, A. O. L.; Li, Wen-Ching Winnie; Long, Ling

On Atkin and Swinnerton-Dyer Congruence Relations (2)

Authors

Atkin, A. O. L.

Li, Wen-Ching Winnie

Long, Ling

Description

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the $p$-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigen forms. We also show that there is an automorphic $L$-function over $\mathbb Q$ whose local factors agree with those of the $l$-adic Scholl representations attached to the space of noncongruence cusp forms.
Last version, to appear on Math Annalen