Vector-valued Automorphic Forms and Vector Bundles

Abstract

In this thesis we prove the existence of vector-valued automorphic forms for an arbitrary Fuchsian group and an arbitrary finite dimensional complex representation of this group. For small enough values of the weight as well as for large enough values, we provide explicit formulas for the spaces of these vector-valued automorphic forms (holomorphic and cuspidal).

To achieve these results, we realize vector-valued automorphic forms as global sections of a certain family of holomorphic vector bundles on a certain Riemann surface associated to the Fuchsian group. The dimension formulas are then provided by the Riemann-Roch theorem.

In the cases of 1 and 2-dimensional representations, we give some applications to the theories of generalized automorphic forms and equivariant functions.