Quantization and Arithmetic

ISBN-13: 9783764387907 / Angielski / Miękka / 2008 / 160 str.

Andre Unterberger; Andr Unterberger

Opis

(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"

Spis treści

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