Trace formulae and applications to class numbers

• Autorzy: Nicole Raulf
• Języki publikacji: EN

Abstrakty

EN

In this paper we compute the trace formula for Hecke operators acting on automorphic forms on the hyperbolic 3-space for the group PSL2($\mathcal{O}_K $) with $\mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field K of class number H K > 1. Furthermore, as a corollary we obtain an asymptotic result for class numbers of binary quadratic forms.

Słowa kluczowe

EN

Hecke operators; Trace formula; Class numbers

Kategorie tematyczne

• 11F72: Spectral theory; Selberg trace formula
• 11F25: Hecke-Petersson operators, differential operators (one variable)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 6
• Strony: 824-847

Daty

wydano

2014-06-01

online

2014-03-19

Twórcy

autor

Nicole Raulf

• Université Lille 1

Bibliografia

• [1] Bauer P., Zeta functions for equivalence classes of binary quadratic forms, Proc. London Math. Soc., 1994, 69(2), 250–276 http://dx.doi.org/10.1112/plms/s3-69.2.250
• [2] Bauer-Price P., The Selberg Trace Formula for PSL(2, \(\mathcal{O}_K \) ) for Imaginary Quadratic Number Fields K of Arbitrary Class Number, Bonner Math. Schriften, 221, Universität Bonn, 1991
• [3] Deitmar A., Hoffmann W., Asymptotics of class numbers, Invent. Math., 2005, 160(3), 647–675 http://dx.doi.org/10.1007/s00222-004-0423-y
• [4] Efrat I.Y., The Selberg Trace Formula for PSL2(ℝ)n, Mem. Amer. Math. Soc., 65(359), American Mathematical Society, Providence, 1987
• [5] Elstrodt J., Grunewald F., Mennicke J., Groups Acting on Hyperbolic Space, Springer Monogr. Math., Springer, Berlin, 1998 http://dx.doi.org/10.1007/978-3-662-03626-6
• [6] Gauß C.F., Untersuchungen über Höhere Arithmetik, Chelsea, New York, 1965
• [7] Heitkamp D., Hecke-Theorie zur SL(2, \(\mathfrak{O}\) ), PhD thesis, Universität Münster, 1990
• [8] Hejhal D.A., The Selberg Trace Formula for PSL(2,ℝ), I, Lecture Notes in Math., 548, Springer, Berlin-New York, 1976
• [9] Hejhal D.A., The Selberg Trace Formula for PSL(2,ℝ), II, Lecture Notes in Math., 1001, Springer, Berlin, 1983
• [10] Iwaniec H., Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Revista Matemática Iberoamericana, Madrid, 1995
• [11] Lang S., Algebraic Number Theory, 2nd ed., Grad. Texts in Math., 110, Springer, New York, 1994 http://dx.doi.org/10.1007/978-1-4612-0853-2
• [12] Li X.-L., On the trace of Hecke operators for Maass forms, In: Number Theory, Ottawa, August 17–22, 1996, CRM Proc. Lecture Notes, 19, American Mathematical Society, Providence, 1999, 215–229
• [13] Neukirch J., Algebraische Zahlentheorie, Springer, Berlin, 2007
• [14] Raulf N., Traces of Hecke Operators Acting on Three-Dimensional Hyperbolic Space, PhD thesis, Universität Münster, 2004
• [15] Raulf N., Traces of Hecke operators acting on three-dimensional hyperbolic space, J. Reine Angew. Math., 2006, 591, 111–148
• [16] Sarnak P., Class numbers of indefinite binary quadratic forms, J. Number Theory, 1982, 15(2), 229–247 http://dx.doi.org/10.1016/0022-314X(82)90028-2
• [17] Sarnak P., The arithmetic and geometry of some hyperbolic three manifolds, Acta Math., 1983, 151(3–4), 253–295 http://dx.doi.org/10.1007/BF02393209
• [18] Selberg A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), 1956, 20, 47–87
• [19] Selberg A., Harmonic analysis, In: Collected Papers, I, Springer, Berlin, 1989
• [20] Serre J.-P., Répartition asymptotique des valeurs propres de l’opérateur de Hecke T p, J. Amer. Math. Soc., 1997, 10(1), 75–102 http://dx.doi.org/10.1090/S0894-0347-97-00220-8
• [21] Siegel C.L., The average measure of quadratic forms with given determinant and signature, Ann. of Math., 1944, 45, 667–685 http://dx.doi.org/10.2307/1969296
• [22] Speiser A., Die Theorie der Binären Quadratischen Formen mit Koeffizienten und Unbestimmten in einem Beliebigen Zahlkörper, PhD thesis, Göttingen, 1909
• [23] Tenenbaum G., Introduction à la Théorie Analytique et Probabiliste des Nombres, 2nd ed., Cours Spec., 1, Société Mathématique de France, Paris, 1995
• [24] Zagier D., The Eichler-Selberg trace formula on SL2(ℤ), Appendix to: Lang S., Introduction to Modular Forms, Grundlehren Math. Wiss., 222, Springer, Berlin-New York, 1976, 44–54

Identyfikatory

DOI

10.2478/s11533-013-0384-8