On Atkin and Swinnerton-Dyer Congruence Relations (2)

Details On Atkin and Swinnerton-Dyer Congruence Relations (2) On Atkin and Swinnerton-Dyer Congruence Relations (2)

2005-12-28

arxiv.org/abs/math/0512614v3

Mathematische Annalen, Volume 340 No. 2 (2008), pp 335-358

Mathematics

Number Theory

Last version, to appear on Math Annalen

Scientific paper

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.

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