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Automorphic forms on the upper half plane have been studied for a long time. He extended Hecke’s relation between automorphic forms and Dirichlet series to real analytic automorphic forms.
Modular introduction.- Modular introduction.- General theory.- Automorphic forms on the universal covering group.- Discrete subgroups.- Automorphic forms.- Poincaré series.- Selfadjoint extension of the Casimir operator.- Families of automorphic forms.- Transformation and truncation.- Pseudo Casimir operator.- Meromorphic continuation of Poincaré series.- Poincaré families along vertical lines.- Singularities of Poincaré families.- Examples.- Automorphic forms for the modular group.- Automorphic forms for the theta group.- Automorphic forms for the commutator subgroup.
From reviews: "It is made abundantly clear that this viewpoint, of families of automorphic functions depending on varying eigenvalue and multiplier systems, is both deep and fruitful." - MathSciNet