# Quantization and Arithmetic [electronic resource] / by André Unterberger.

##### By: Unterberger, André [author.].

##### Contributor(s): SpringerLink (Online service).

Material type: TextSeries: Pseudo-Differential Operators, Theory and Applications: 1Publisher: Basel : Birkhäuser Basel, 2008Description: 147 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783764387914.Subject(s): Topological Groups | Number theory | Mathematical physics | Operator theory | Topological Groups, Lie Groups | Number Theory | Mathematical Methods in Physics | Operator TheoryAdditional physical formats: Printed edition:: No title; Printed edition:: No titleDDC classification: 512.55 | 512.482 Online resources: Click here to access onlineWeyl Calculus and Arithmetic -- Quantization -- Quantization and Modular Forms -- Back to the Weyl Calculus.

(12) (4) Let ? be the unique even non-trivial Dirichlet character mod 12, and let ? be the unique (odd) non-trivial Dirichlet character mod 4. Consider on the line the distributions m (12) ? d (x)= ? (m)? x? , even 12 m?Z m (4) d (x)= ? (m)? x? . (1.1) odd 2 m?Z 2 i?x UnderaFouriertransformation,orundermultiplicationbythefunctionx ? e , the?rst(resp. second)ofthesedistributionsonlyundergoesmultiplicationbysome 24th (resp. 8th) root of unity. Then, consider the metaplectic representation Met, 2 a unitary representation in L (R) of the metaplectic group G, the twofold cover of the group G = SL(2,R), the de?nition of which will be recalled in Section 2: it extends as a representation in the spaceS (R) of tempered distributions. From what has just been said, if g ˜ is a point of G lying above g? G,andif d = d even g ˜ ?1 or d , the distribution d =Met(g˜ )d only depends on the class of g in the odd homogeneousspace?\G=SL(2,Z)\G,uptomultiplicationbysomephasefactor, by which we mean any complex number of absolute value 1 depending only on g ˜. On the other hand, a function u?S(R) is perfectly characterized by its scalar g ˜ productsagainstthedistributionsd ,sinceonehasforsomeappropriateconstants C , C the identities 0 1 g ˜ 2 2 | d ,u | dg = C u if u is even, 2 0 even L (R) ?\G.

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