Quantization and Arithmetic
Seiten
2008
|
2008
Springer Basel (Verlag)
978-3-7643-8790-7 (ISBN)
Springer Basel (Verlag)
978-3-7643-8790-7 (ISBN)
This book creates situations in which the zeta function, or other L-functions, appear in spectral-theoretic questions. It also connects pseudo-differential analysis, or quantization theory, to analytic number theory.
(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
(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
Weyl Calculus and Arithmetic.- Quantization.- Quantization and Modular Forms.- Back to the Weyl Calculus.
| Erscheint lt. Verlag | 23.6.2008 |
|---|---|
| Reihe/Serie | Pseudo-Differential Operators |
| Zusatzinfo | 147 p. |
| Verlagsort | Basel |
| Sprache | englisch |
| Maße | 170 x 240 mm |
| Gewicht | 395 g |
| Themenwelt | Mathematik / Informatik ► Mathematik |
| Schlagworte | Calculus • Coherent states • Hardcover, Softcover / Mathematik/Arithmetik, Algebra • HC/Mathematik/Arithmetik, Algebra • Matrix • Metaplectic representation • Number Theory • Quantization • Weyl calculus |
| ISBN-10 | 3-7643-8790-4 / 3764387904 |
| ISBN-13 | 978-3-7643-8790-7 / 9783764387907 |
| Zustand | Neuware |
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