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Number theory --- Automorphic forms. --- Fourier series. --- Nombres, Théorie des --- Formes automorphes --- Automorphic forms --- Number theory. --- Nombres, Théories des --- Fourier, Séries de

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Topological groups. Lie groups --- Lie groups. --- Representations of groups. --- Groupes de Lie --- Représentations de groupes --- 51 <082.1> --- Mathematics--Series --- Représentations de groupes --- Lie, Groupes de. --- Représentations de groupes. --- Lie groups --- Representations of groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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This book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the “abelian” Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the “non-abelian” modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included. These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms. Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.

Number theory. --- Fourier analysis. --- Topological groups. --- Lie groups. --- Number Theory. --- Fourier Analysis. --- Topological Groups and Lie Groups. --- Teoria de nombres --- Anàlisi de Fourier

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Number theory --- Ordered algebraic structures --- Topological groups. Lie groups --- Harmonic analysis. Fourier analysis --- Fourieranalyse --- wiskunde --- getallenleer --- topologie

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The authors investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. The authors use Knopp's generalization of this integral to real weights, and apply it to complex weights that are not an integer at least 2. They show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. The authors impose no condition on the growth of the automorphic forms at the cusps. Their result concerns arbitrary cofinite discrete groups with cusps, and covers exponentially growing automorphic forms, like those studied by Borcherds, and like those in the theory of mock automorphic forms.

Automorphic forms. --- Holomorphic functions. --- Cohomology operations.