Mathematische Annalen

, Volume 340, Issue 2, pp 335–358

On Atkin and Swinnerton-Dyer congruence relations (2)

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

DOI: 10.1007/s00208-007-0154-7

Cite this article as:
Atkin, A.O.L., Li, WC.W. & Long, L. Math. Ann. (2008) 340: 335. doi:10.1007/s00208-007-0154-7


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 eigenforms. 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.

Mathematics Subject Classification (2000)

11F11 11F33 

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • A. O. L. Atkin
    • 1
  • Wen-Ching Winnie Li
    • 2
  • Ling Long
    • 3
  1. 1.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  3. 3.Department of MathematicsIowa State UniversityAmesUSA

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