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Erschienen in:
2023 | OriginalPaper | Buchkapitel
5. Application to Automorphic Forms
verfasst von : Roelof W. Bruggeman, Roberto J. Miatello
Erschienen in: Representations of SU(2,1) in Fourier Term Modules
Verlag: Springer Nature Switzerland
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Abstract
Finally, we use the knowledge concerning Fourier term modules to understand better the Fourier expansion of automorphic forms.
Usually, automorphic forms are required to have at most polynomial growth at the cusps. Here we also define automorphic forms with moderate exponential growth. A growth condition on the modular form implies properties of the Fourier expansion. For \({\mathrm {SL}}_2(\mathbb {R})\), an automorphic form with Fourier terms that have polynomial growth has polynomial growth itself. For \({\mathrm {SU}}(2,1)\) this does not necessarily hold.
We consider also the Fourier expansion of families of automorphic forms, and of generating vectors of irreducible automorphic modules.