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Birkhäuser

Quantization and Arithmetic

  • Book
  • © 2008

Overview

Authors:
  1. André Unterberger

    1. Département Mathématiques et Informatique, Université de Reims, Reims Cedex 2, France

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  • Consists entirely of new material
  • Contents is related to the book Automorphic pseudodifferential analysis and higher-level Weyl calculi (appeared in Progress in Mathematics) by the same author: one-dimensional pseudo-differential analysis is considered in both volumes; however, the results are disjoint: the former book dealt with automorphic symbols, while in the present one, it is the distributions to which the operators under consideration are applied that carry the arithmetic; in this way, L-functions, rather than non-holomorphic modular forms, have the dominant role
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (5 chapters)

  1. Front Matter

    Pages I-V
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  2. Introduction

    Pages 1-6

  3. Weyl Calculus and Arithmetic

    Pages 7-47

  4. Quantization

    Pages 49-70

  5. Quantization and Modular Forms

    Pages 71-116

  6. Back to the Weyl Calculus

    Pages 117-141

  7. Back Matter

    Pages 143-147
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Keywords

About this book

(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

Authors and Affiliations

  • Département Mathématiques et Informatique, Université de Reims, Reims Cedex 2, France

    André Unterberger

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