Quantization and non-holomorphic modular forms. (English) Zbl 0970.11014

Unterberger, André

Lecture Notes in Mathematics. 1742. Berlin: Springer. viii, 253 p. (2000).

Although the derivative of a modular form is not a modular form in general, certain combinations of derivatives of modular forms produce modular forms. Indeed, the polynomials in the derivatives of modular forms for a discrete subgroup \(\Gamma\) of \(\text{SL}(2,\mathbb R)\) that are again modular forms were studied by R. J. Rankin [J. Indian Math. Soc., New Ser. 20, 103–116 (1956; Zbl 0072.08601)], and as a special case of such polynomials H. Cohen [Math. Ann. 217, 271–285 (1975; Zbl 0311.10030)] investigated certain bilinear operators on the graded ring of modular forms, which may be considered as noncommutative products of modular forms. Such products are known as the Rankin-Cohen brackets, and they can be extended to the case of Hilbert or Siegel modular forms. They are also closely linked to various objects including pseudodifferential operators, Jacobi-like forms, and transvectants.
One of the main goals of this monograph is to generalize the Rankin-Cohen brackets of modular forms to the case of non-holomorphic automorphic forms. In fact, the author envisions the non-holomorphic analogue of Rankin-Cohen brackets as a machine for producing Maass cusp forms. In the process of constructing the bilinear products, which generalize the Rankin-Cohen brackets, the author uses various techniques from pseudodifferential analysis, partial differential equations, and harmonic analysis such as the Radon transform, the Rankin-Selberg unfolding method, Weyl symbols, and Poisson brackets. In addition he discusses connections of such bilinear products with quantization theory.

Reviewer: Min Ho Lee (Cedar Falls)

MSC:

11F11	Holomorphic modular forms of integral weight
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11F25	Hecke-Petersson operators, differential operators (one variable)
11L05	Gauss and Kloosterman sums; generalizations
44A12	Radon transform
81S99	General quantum mechanics and problems of quantization
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35S99	Pseudodifferential operators and other generalizations of partial differential operators

Keywords:

non-holomorphic modular forms; Maass cusp forms; Eisenstein series; Rankin-Cohen brackets; Weyl symbols; Poisson brackets; Radon transform; connections of bilinear products with quantization theory

Citations:

Zbl 0072.08601; Zbl 0311.10030