Pseudo-Differential Operators

Quantization and Arithmetic

Authors: Unterberger, André

  • 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 deals with automorphic symbols, while in this 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
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(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

Table of contents (5 chapters)

Buy this book

eBook $54.99
price for USA (gross)
  • ISBN 978-3-7643-8791-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $69.99
price for USA
  • ISBN 978-3-7643-8790-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Quantization and Arithmetic
Authors
Series Title
Pseudo-Differential Operators
Series Volume
1
Copyright
2008
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Basel
eBook ISBN
978-3-7643-8791-4
DOI
10.1007/978-3-7643-8791-4
Softcover ISBN
978-3-7643-8790-7
Series ISSN
2297-0355
Edition Number
1
Number of Pages
147
Topics