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

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.

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 |