Modular Forms on Noncongruence Subgroups and Atkin-Swinnerton-Dyer Relations

We give an example of a noncongruence subgroup Γ ⊂ SL(2, Z) whose space of weight 3 cusp forms S3(Γ) admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with respect to a weight 3 newform for a certain congruence subgroup. This gives a modularity interpretation of the motive attached to S3(Γ) by A. Scholl and also verifies the Atkin-Swinnerton-Dyer congruence conjecture for this space.

• Type: Journal
• Year: 2010
• Where: EM
• Authors: Liqun Fang, J. William Hoffman, Benjamin Linowitz, Andrew Rupinski, Helena A. Verrill
• Keywords: Atkin-Swinnerton-Dyer Congruence, Atkin-Swinnerton-Dyer Congruence Relations, Certain Congruence Subgroup, EM 2010, Management