Imbedded modular curves and arithmetic of automorphic forms on bounded symmetric domains
type: Item
instance of: scholarly article
title: Imbedded modular curves and arithmetic of automorphic forms on bounded symmetric domains
zbMATH Open document ID: 0545.10019
DOI: 10.1215/S0012-7094-84-05122-6
author: Paul Garrett
published in: Duke Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The author introduces three different notions of ''arithmeticity'' of (holomorphic) automorphic forms on (irreducible) bounded symmetric domains \(\tilde D\) (with noncompact quotients) namely, via 1) ''special values'' (of automorphic forms) at ''CM-points'', 2) ''pullbacks'' to elliptic modular forms and 3) Fourier-Jacobi coefficients and proves the equivalence of the three definitions. Relations between specializations of ''normalized division-point values'' (generalizing the traditional Fourier coefficients) of automorphic forms (on an ''unbounded realization'' \(\tilde D_ s)\) and the Fourier coefficients of the elliptic modular form obtained by ''pullback'' are studied in a detailed manner [see the author, Am. J. Math. 105, 1171-1193 and 1195-1216 (1983; Zbl 0527.10024 and Zbl 0527.10025)]; not only does the collection of ''pullbacks'' recover the original automorphic form but ''sufficient arithmetic information'' about the form is encoded therein as well. The spaces of ''arithmetic'' automorphic forms on \(\tilde D\) turn out to be stable under the translation-action of the rational points of the associated group (of holomorphic isometries of \~D); the phenomenon of such spaces going over into themselves under the ''usual Hecke operators'' is a Corollary. ''Arithmeticity'' is shown to be preserved under ''pullbacks'' via equivariant morphisms between domains. After comparing the definition of ''arithmeticity'' for various unbounded realizations of \(\tilde D\) the author finally shows that an automorphic form on any such \(\tilde D\) (with a zero-dimensional rational boundary component) is ''arithmetic'' if and only if its ''normalized division-point values'' are ''arithmetic'' automorphic forms on rational boundary components of \~D.
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 32N15
zbMATH DE Number: 3867430
zbMATH Keywords: arithmetic automorphic form
zbMATH Keywords: special values at CM-points
zbMATH Keywords: bounded symmetric domains
zbMATH Keywords: pullbacks
zbMATH Keywords: elliptic modular forms
zbMATH Keywords: Fourier-Jacobi coefficients
zbMATH Keywords: equivalence
zbMATH Keywords: normalized division-point values
MaRDI profile type: Publication
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full work available at URL: s0012-7094-84-05122-6
OpenAlex ID: W1551359465
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description: scientific article; zbMATH DE number 3867430