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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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  • [dB] N. G. de Bruijn, Over modulaire vormen van meer veranderlijken, Thesis, Free University of Amsterdam, 1943 MR 0016387 (8:8d)
  • [G1] K.-B. Gundlach, Poincaresche und Eisensteinsche Reihen zur Hilbertschen Modulgruppe, Math. Z. 64 (1956), 339-352. MR 0080124 (18:195d)
  • [G2] -, Dirichletsche Reihen zur Hilbertschen Modulgruppe, Math. Ann. 135 (1958), 294-314. MR 0104642 (21:3395)
  • [H] O. Herrmann, Uber Hilbertsche Modulfunctionen und die Dirichletschen Reihen mit Eulerschen Producktentwicklung, Math. Ann. 127 (1954), 357-400. MR 0062181 (15:940d)
  • [La] S. Lang, Algebraic number theory, Addison-Wesley, New York, 1970. MR 0282947 (44:181)
  • [R] W. Roelke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II, Math. Ann. 167 (1966), 292-337; 168 (1967), 261-324. MR 1513277
  • [Sa] P. Sarnak, The arithmetic and geometry of some hyperbolic three manifolds, Acta Math. 151 (1984), 253-295. MR 723012 (85d:11061)
  • [Se] J. P. Serre, Trees, Springer-Verlag, Berlin and New York, 1980. MR 607504 (82c:20083)
  • [Sc] A. Schwartz, Modular forms of weight $ \frac{1} {2}$ defined on products of $ p$-adic upper half-planes, Trans. Amer. Math. Soc. 335 (1993), 757-773.
  • [Si] C. L. Siegel, Lectures on advanced analytic number theory, Tata Institute Lectures, Bombay, 1961. MR 0262150 (41:6760)
  • [Sh] G. Shimura, Arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, NJ, 1971. MR 0314766 (47:3318)
  • [St] H. M. Stark, Modular forms and related objects, Number Theory (Proc. 1985 Montreal Conf.), Amer. Math. Soc., Providence, RI, 1987, pp. 421-455. MR 894333 (88j:11029)
  • [Sr] R. Styer, Hecke theory over arbitrary number fields, J. Number Theory 33 (1989), 107-131. MR 1034194 (91a:11020)
  • [T] J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, Algebraic Number Theory, Thompson, Washington, DC, 1967, pp. 305-347. MR 0217026 (36:121)
  • [Te] A. Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406 (87f:22010)
  • [W] A. Weil, Dirichlet series and automorphic forms, Lecture Notes in Math., vol. 189, Springer-Verlag, Berlin and New York, 1971.

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