On Atkin and Swinnerton-Dyer Congruence Relations (2)

TL;DR

This paper provides an example of a noncongruence subgroup with a three-dimensional space of cusp forms of weight 3, demonstrating Atkin and Swinnerton-Dyer congruence relations and linking to automorphic L-functions.

Contribution

It introduces a specific noncongruence subgroup with cusp forms satisfying p-adic Atkin and Swinnerton-Dyer relations and connects these forms to automorphic L-functions over Q.

Findings

Existence of a noncongruence subgroup with specific cusp form properties

Four residue classes of primes exhibit Atkin and Swinnerton-Dyer congruences

Automorphic L-function matches local factors of Scholl representations

Abstract

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.