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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

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Harmonic differentials and infinite geodesic joining two punctures on a Riemann surface
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by Thérèse Falliero
Trans. Amer. Math. Soc. 363 (2011), 3473-3488
DOI: https://doi.org/10.1090/S0002-9947-2011-05224-4
Published electronically: February 10, 2011
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Abstract:

Let $M$ be a hyperbolic Riemann surface of finite volume. The harmonic dual form to an infinite geodesic joining two punctures on $M$ is obtained in two different ways. First of all, using the degeneration of hyperbolic Eisenstein series, it is made explicit in terms of these. Secondly, generalizing the construction of Kudla and Millson to the case of an infinite geodesic joining two punctures, we give an automorphic realization of this harmonic form.
References
  • Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
  • Thérèse Falliero, Décomposition spectrale de 1-formes différentielles sur une surface de Riemann et séries d’Eisenstein, Math. Ann. 317 (2000), no. 2, 263–284 (French, with English and French summaries). MR 1764237, DOI 10.1007/s002080000093
  • Thérèse Falliero, Dégénérescence de séries d’Eisenstein hyperboliques, Math. Ann. 339 (2007), no. 2, 341–375 (French, with English and French summaries). MR 2324723, DOI 10.1007/s00208-007-0116-0
  • John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
  • John D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293(294) (1977), 143–203. MR 506038, DOI 10.1515/crll.1977.293-294.143
  • P. Gérardin, Formes automorphes associées aux cycles géodésiques des surfaces de Riemann hyperboliques, Séminaire Bourbaki 562 (1980).
  • G. Harder, On the cohomology of $SL(2,O)$, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 139–150. MR 0425019
  • G. Harder, On the cohomology of discrete arithmetically defined groups, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 129–160. MR 0425018
  • Dennis A. Hejhal, Theta functions, kernel functions, and Abelian integrals, Memoirs of the American Mathematical Society, No. 129, American Mathematical Society, Providence, R.I., 1972. MR 0372187
  • Dennis A. Hejhal, A continuity method for spectral theory on Fuchsian groups, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 107–140. MR 803365
  • Dennis A. Hejhal, Regular $b$-groups, degenerating Riemann surfaces, and spectral theory, Mem. Amer. Math. Soc. 88 (1990), no. 437, iv+138. MR 1052555, DOI 10.1090/memo/0437
  • Henryk Iwaniec, Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matemática Iberoamericana. [Library of the Revista Matemática Iberoamericana], Revista Matemática Iberoamericana, Madrid, 1995. MR 1325466
  • Lizhen Ji, Spectral degeneration of hyperbolic Riemann surfaces, J. Differential Geom. 38 (1993), no. 2, 263–313. MR 1237486
  • Lizhen Ji, The asymptotic behavior of Green’s functions for degenerating hyperbolic surfaces, Math. Z. 212 (1993), no. 3, 375–394. MR 1207299, DOI 10.1007/BF02571664
  • Jay Jorgenson, Asymptotic behavior of Faltings’s delta function, Duke Math. J. 61 (1990), no. 1, 221–254. MR 1068387, DOI 10.1215/S0012-7094-90-06111-3
  • Stephen S. Kudla and John J. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979), no. 3, 193–211. MR 553218, DOI 10.1007/BF01390229
  • Stephen S. Kudla and John J. Millson, The Poincaré dual of a geodesic algebraic curve in a quotient of the $2$-ball, Canadian J. Math. 33 (1981), no. 2, 485–499. MR 617638, DOI 10.4153/CJM-1981-042-x
  • Stephen S. Kudla and John J. Millson, Geodesic cycles and the Weil representation. I. Quotients of hyperbolic space and Siegel modular forms, Compositio Math. 45 (1982), no. 2, 207–271. MR 651982
  • Aaron Lebowitz, On the degeneration of Riemann surfaces, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 265–286. MR 0281908
  • Barry Mazur, Courbes elliptiques et symboles modulaires, Séminaire Bourbaki, 24ème année (1971/1972), Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. Exp. No. 414, pp. 277–294. MR 0429921
  • David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651, DOI 10.1007/978-1-4899-2843-6
  • Kunio Obitsu, Non-completeness of Zograf-Takhtajan’s Kähler metric for Teichmüller space of punctured Riemann surfaces, Comm. Math. Phys. 205 (1999), no. 2, 405–420. MR 1712579, DOI 10.1007/s002200050683
  • Kunio Obitsu, The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics, Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, 459–478. MR 1855431
  • Kunio Obitsu and Scott A. Wolpert, Grafting hyperbolic metrics and Eisenstein series, Math. Ann. 341 (2008), no. 3, 685–706. MR 2399166, DOI 10.1007/s00208-008-0210-y
  • K. Obitsu, Asymptotics of degenerating Eisenstein series. Arxiv:0801.3691(2008).
  • R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $\textrm {PSL}(2,\textbf {R})$, Invent. Math. 80 (1985), no. 2, 339–364. MR 788414, DOI 10.1007/BF01388610
  • Hans Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman. MR 0364103
  • C. L. Siegel, Topics in complex function theory, Tracts in Mathematics Number 25, vol. 2, Wiley-Interscience, 1971.
  • R. Wentworth, The asymptotics of the Arakelov-Green’s function and Faltings’ delta invariant, Comm. Math. Phys. 137 (1991), no. 3, 427–459. MR 1105425
  • Scott A. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 2, 283–315. MR 905169
  • Scott A. Wolpert, Spectral limits for hyperbolic surfaces. I, II, Invent. Math. 108 (1992), no. 1, 67–89, 91–129. MR 1156387, DOI 10.1007/BF02100600
  • Scott A. Wolpert, Spectral limits for hyperbolic surfaces. I, II, Invent. Math. 108 (1992), no. 1, 67–89, 91–129. MR 1156387, DOI 10.1007/BF02100600
  • Scott A. Wolpert, Disappearance of cusp forms in special families, Ann. of Math. (2) 139 (1994), no. 2, 239–291. MR 1274093, DOI 10.2307/2946582
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Bibliographic Information
  • Thérèse Falliero
  • Affiliation: Laboratoire d’analyse non linéaire et géométrie (E-A 251), Université d’Avignon et des Pays de Vaucluse, F-84018 Avignon, France
  • Email: therese.falliero@univ-avignon.fr
  • Received by editor(s): February 20, 2009
  • Published electronically: February 10, 2011
  • © Copyright 2011 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 3473-3488
  • MSC (2010): Primary 53C20, 53C22, 58A10; Secondary 58D27
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05224-4
  • MathSciNet review: 2775815