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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Differential operators and theta series

Author: Solomon Friedberg
Journal: Trans. Amer. Math. Soc. 287 (1985), 569-589
MSC: Primary 11F25; Secondary 11F11
MathSciNet review: 768726
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Abstract: Let $ f$ be a modular form on a congruence subgroup of $ {\text{SL}}(2,\mathbb{Z})$--not necessarily holomorphic, but an eigenfunction of the weight $ k$ Casimir operator. Maass introduced differential operators (coming from the complexified universal enveloping algebra) which raise and lower by $ 2$ the weight of such a form and shift the eigenvalue. Here we introduce differential operators on hyperbolic $ 3$ space analogous to the Maass operators. These change by $ 2$ 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:

\begin{displaymath}\begin{array}{*{20}{c}} & F & {\underset{{{\text{operators}}}... ...ext{Maass}}}}{\leftrightarrow}}} & {\tilde f} & \\ \end{array} \end{displaymath}

Using similar techniques for the dual pair $ ({\text{SL}}(2,\mathbb{R}),\;{\text{SO}}(2,1))$, we give a simple proof that the Shimura correspondences preserve holomorphicity (for weight $ \geqslant 5/2$) 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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Keywords: Theta function, modular form, Doi-Naganuma lifting, Shimura correspondence, Maass operator, hyperbolic $ 3$-space
Article copyright: © Copyright 1985 American Mathematical Society