Quantization and Arithmetic

(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

List of contents

Weyl Calculus and Arithmetic.- Quantization.- Quantization and Modular Forms.- Back to the Weyl Calculus.

Product details

• Authors: André Unterberger
• Publisher: Springer, Basel
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2008
• EAN: 9783764387907
• ISBN: 978-3-7643-8790-7
• No. of pages: 160
• Dimensions: 170 mm x 8 mm x 240 mm
• Weight: 284 g
• Illustrations: 147 p.
• Series: Pseudo-differential Operators
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; B, Mathematische Physik, Funktionalanalysis und Abwandlungen, Physics, Mathematics and Statistics, Number Theory, Mathematical physics, Topological Groups, Lie Groups, Topological groups, Lie groups, Topological Groups and Lie Groups, Mathematical Methods in Physics, Functional analysis & transforms, Operator Theory