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Quantization and Modular Forms

Part of the Pseudo-Differential Operators book series (PDO, volume 1)

Abstract

We here extend the analysis of operators by means of their diagonal matrix elements, this time against a family of arithmetic coherent states adapted to the representation \( \mathcal{D}_{\tau + 1} \). The basic distribution \( \mathfrak{s}_\tau \) they are built from, substituting for the distributions δeven and δodd of Chapter 1, is just another realization of a modular form f of weight τ + 1 of some kind. Section 9 describes some possibilities: one may for instance consider a power of the Dedekind eta-function. The distributions \( \mathfrak{s}_\tau ^{\tilde g} \), where \( \tilde g \) lies in some homogeneous space of the universal cover of G above Γ\G, then constitute a family of (arithmetic) coherent states for the representation πτ+1 in the way described by a formula of resolution of the identity analogous to (1.2): note that the existence of such a formula depends in a crucial way on the fact that f is a cusp-form.

Keywords

Meromorphic Function Modular Form Wigner Function Fundamental Domain Eisenstein Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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