Imbedded modular curves and arithmetic of automorphic forms on bounded symmetric domains
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.
scientific article; zbMATH DE number 3867430
scientific article