Quantization and arithmetic
著者
書誌事項
Quantization and arithmetic
（Pseudo-differential operators : theory and applications / managing editor, M.W. Wong, v. 1）
Birkhäuser, c2008
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注記
Includes bibliographical references (p. [145]-147) and index
内容説明・目次
内容説明
(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
目次
Weyl Calculus and Arithmetic.- Quantization.- Quantization and Modular Forms.- Back to the Weyl Calculus.
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