Differential operators and theta series

Author:
Solomon Friedberg

Journal:
Trans. Amer. Math. Soc. **287** (1985), 569-589

MSC:
Primary 11F25; Secondary 11F11

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768726-4

MathSciNet review:
768726

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a modular form on a congruence subgroup of --not necessarily holomorphic, but an eigenfunction of the weight Casimir operator. Maass introduced differential operators (coming from the complexified universal enveloping algebra) which raise and lower by the weight of such a form and shift the eigenvalue. Here we introduce differential operators *on hyperbolic* *space* analogous to the Maass operators. These change by the weight of a modular form for an *imaginary quadratic field*.

Theorem. *The Maass operators and the hyperbolic space operators are intertwined by the imaginary quadratic Doi-Naganuma (base change) lifting. That is, the following diagram is commutative*:

Using similar techniques for the dual pair , we give a simple proof that the Shimura correspondences preserve holomorphicity (for weight ) and an explanation for this property directly in terms of the theta series (Weil representation) integral kernel. We also establish similar results for the real quadratic Doi-Naganuma lifting.

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Goro Shimura,*On certain zeta functions attached to two Hilbert modular forms. II. The case of automorphic forms on a quaternion algebra*, Ann. of Math. (2)**114**(1981), no. 3, 569–607. MR**634429**, https://doi.org/10.2307/1971302**[13]**Takuro Shintani,*On construction of holomorphic cusp forms of half integral weight*, Nagoya Math. J.**58**(1975), 83–126. MR**0389772****[14]**A. Weil,*Dirichlet series and automorphic forms*, Lecture Notes in Math., vol. 189, Springer, 1971.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768726-4

Keywords:
Theta function,
modular form,
Doi-Naganuma lifting,
Shimura correspondence,
Maass operator,
hyperbolic -space

Article copyright:
© Copyright 1985
American Mathematical Society