• DOI:10.1090/S0002-9939-2011-11045-1
• Corpus ID: 27594101

On ℓ-adic representations for a space of noncongruence cuspforms

@inproceedings{Hoffman2010OnR,
  title={On ℓ-adic representations for a space of noncongruence cuspforms},
  author={J. William Hoffman and Ling Long and Helena A. Verrill},
  year={2010}
}

• J. W. Hoffman, L. Long, H. Verrill
• Published 19 March 2010
• Mathematics

This paper is concerned with a compatible family of 4-dimensional l-adic representations ρl of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that:

1.
It is automorphic: the L-function L(s,ρl∨) agrees with the L-function for an automorphic form for GL4(AQ), where ρl∨ is the dual of ρl.
2.
For each prime p≥5 there is a basis hp = {hp+, hp-} of S3(Γ) whose expansion coefficients satisfy 3-term Atkin and Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a newform f of level 432. The structure of this basis depends on the class of p modulo 12.
The key point is that the representation ρl admits a quaternion multiplication structure in the sense of Atkin, Li, Liu, and Long.