A Hecke correspondence theorem for automorphic integrals with infinite log-polynomial sum period functions

We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods.

EICHLER COHOMOLOGY THEOREM FOR GENERALIZED MODULAR FORMS

We show starting with relations between Fourier coefficients of weakly parabolic generalized modular forms of negative weight that we can construct automorphic integrals for large integer weights. We finally prove an Eichler isomorphism theorem for weakly parabolic generalized modular forms using the classical approach as in [3].