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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] Nicolaas Govert de Bruijn, Over Modulaire Vormen van Meer Veranderlijken, Thesis, Free University of Amsterdam,], 1943 (Dutch). MR 0016387
  • [G1] Karl-Bernhard Gundlach, Poincáresche und Eisensteinsche Reihen zur Hilbertschen Modulgruppe, Math. Z. 64 (1956), 339–352 (German). MR 0080124
  • [G2] Karl-Bernhard Gundlach, Dirichletsche Reihen zur Hilbertschen Modulgruppe, Math. Ann. 135 (1958), 294–314 (German). MR 0104642
  • [H] Oskar Herrmann, Über Hilbertsche Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, Math. Ann. 127 (1954), 357–400 (German). MR 0062181
  • [La] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [R] Walter Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I, Math. Ann. 167 (1966), no. 4, 292–337 (German). MR 1513277, 10.1007/BF01364540
  • [Sa] P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 3-4, 253–295. MR 723012, 10.1007/BF02393209
  • [Se] Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
  • [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] Carl Ludwig Siegel, Lectures on advanced analytic number theory, Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23, Tata Institute of Fundamental Research, Bombay, 1965. MR 0262150
  • [Sh] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR 0314766
  • [St] H. M. Stark, Modular forms and related objects, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 421–455. MR 894333
  • [Sr] Robert Styer, Hecke theory over arbitrary number fields, J. Number Theory 33 (1989), no. 2, 107–131. MR 1034194, 10.1016/0022-314X(89)90001-2
  • [T] J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347. MR 0217026
  • [Te] Audrey Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406
  • [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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DOI: https://doi.org/10.1090/S0002-9947-1994-1159195-9
Article copyright: © Copyright 1994 American Mathematical Society