On the Dimension of the Space of Cusp Forms Associated to 2-dimensional Complex Galois Representations

The aim of this paper is to use the “amplification technique” to obtain estimates on the dimension of spaces of automorphic forms associated to Galois representations; these bounds improve nontrivially on the work of Duke ([D]). A cuspidal representation π of GL2(AQ) is associated to a 2-dimensional Galois representation ρ : Gal(Q/Q) → GL2(C) if, for each place v, the local representation πv is matched with the induced map ρ : Gal(Qv/Qv)→ GL2(C) under the local Langlands correspondence for GL2(Qv). In particular, if one fixes a central character, there are only two possibilities for π∞. We call a form that is associated to a Galois representation “of Galois type.” More precisely, for G ∈ Types = {Dihedral, Tetrahedral, Octahedral, Icosahedral}, we call a form of “type G” if it associated to a Galois representation whose image in PGL2(C) is dihedral, tetrahedral (A4), octahedral (S4), or icosahedral (A5). Define a real-valued function e on Types via e(Dihedral) = 1/2, e(Tetrahedral) = 2/3, e(Octahedral) = 4/5, and e(Icosahedral) = 6/7. With this definition: