Vector-valued modular functions for the modular group and the hypergeometric equation

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dc.creatorBantay, P.
dc.creatorGannon, T.
dc.date2007-05-17
dc.date.accessioned2026-07-07T08:01:59Z
dc.date.available2026-07-07T08:01:59Z
dc.descriptionA general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary half-integer weight. It is shown that the space of these modular functions is spanned, as a module over the polynomials in J, by the columns of a matrix that satisfies an abstract hypergeometric equation, providing a simple solution of the Riemann-Hilbert problem for representations of the modular group. Restrictions on the coefficients of this differential equation implied by analyticity are discussed, and an inversion formula is presented that allows the determination of an arbitrary vector-valued modular function from its singular behavior. Questions of rationality and positivity of expansion coefficients are addressed. Closed expressions for the number of vector-valued modular forms of half-integer weight are given, and the general theory is illustrated on simple examples.
dc.description20 pages, latex
dc.identifierhttps://arxiv.org/abs/0705.2467
dc.identifierhttp://arxiv.org/abs/0705.2467
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/129093
dc.subjectNumber Theory
dc.subjectHigh Energy Physics - Theory
dc.subject11F03; 81T40, 17B69
dc.titleVector-valued modular functions for the modular group and the hypergeometric equation
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