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Transactions of the American Mathematical Society

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



$ p$-adic hyperbolic planes and modular forms

Author: John A. Rhodes
Journal: Trans. Amer. Math. Soc. 341 (1994), 469-504
MSC: Primary 11F41; Secondary 11F25, 11F85
MathSciNet review: 1159195
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Abstract: For $ K$ a number field and $ {\mathbf{p}}$ a finite prime of $ K$, we define a $ {\mathbf{p}}$-adic hyperbolic plane and study its geometry under the action of $ G{L_2}({K_{\mathbf{p}}})$. Seeking an operator with properties analogous to those of the non-Euclidean Laplacian of the classical hyperbolic plane, we investigate the fundamental invariant integral operator, the Hecke operator $ {T_{\mathbf{p}}}$. Letting $ S$ be a finite set of primes of a totally real $ K$ (including all the infinite ones), a modular group $ \Gamma (S)$ is defined. This group acts discontinuously on a product of classical and $ {\mathbf{p}}$-adic hyperbolic planes. $ S$-modular forms and their associated Dirichlet series are studied.

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