Quantization and Modular Forms

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


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.


Meromorphic Function Modular Form Wigner Function Fundamental Domain Eisenstein Series 
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© Birkhäuser Verlag AG 2008

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