On the residues and special values of Poincaré series

Abstract

Let G be the universal cover of SL(2, R), Γ a lattice in G and χ a unitary character of Γ. In this paper we use residues and special values of the Poincaré series M(ν,χ), for the construction of (χ, Γ)-square integrable automorphic forms on G. In particular, we show that for low weight 1 < r ≤ 2, the special values at ν = r − 1 obtained by meromorphic continuation of M η(ν, χ), as η varies, generate the space of holomorphic cusp forms of weight r and multiplier system υ χ . We also prove a completeness result for holomorphic forms of weight 0 < r ≤ 1 by using residues of the family.