Quantization and Arithmetic

André Unterberger

ISBN 10: 3764387904 / ISBN 13: 9783764387907

Published by Birkhäuser Basel, Springer Basel Jun 2008, 2008

About this Item

Neuware -(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) GSpringer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, 14197 Berlin 160 pp. Englisch.

Seller Inventory # 9783764387907

Bibliographic Details

• Title: Quantization and Arithmetic
• Publisher: Birkhäuser Basel, Springer Basel Jun 2008
• Publication Date: 2008
• Language: English
• ISBN 10: 3764387904
• ISBN 13: 9783764387907
• Binding: Taschenbuch
• Condition: Neu

About this title

Synopsis

The primary aim of this book is to create situations in which the zeta function, or other L-functions, will appear in spectral-theoretic questions. A secondary aim is to connect pseudo-differential analysis, or quantization theory, to analytic number theory. Both are attained through the analysis of operators on functions on the line by means of their diagonal matrix elements against families of arithmetic coherent states: these are families of discretely supported measures on the line, transforming in specific ways under the part of the metaplectic representation or, more generally, representations from the discrete series of SL(2, R), lying above an arithmetic group such as SL(2, Z).