Galois representations with quaternion multiplication associated to noncongruence modular forms

In this paper we study the compatible family of degree-4 Scholl representations ρℓ associated with a space S of weight κ > 2 noncongruence cusp forms satisfying QuaternionMultiplication over a biquadratic extension of ℚ. It is shown that ρℓ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for GL₄ over ℚ. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in S and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.