On a Lemma of Deligne-Serre

This note is intended to provide the group-theoretic machinery for associating complex representations to certain modular forms. We recall [DS] that to a Hecke eigenform of weight 1 on GL2, we can associate a 2-dimensional compatible system of λ-adic representations. Rankin’s method shows that “on average” the eigenvalues of Frobenius are of O(1), and by a group-theoretic argument ([DS] §§7-8), one concludes that the system of representations actually comes from a complex 2-dimensional representation of Gal(Q/Q). We prove the analogous theorem in arbitrary dimension.