On Atkin and Swinnerton-Dyer congruence relations (3) Original Research Article

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of image. Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117–148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335–358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.