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Eisenstein Series and Equidistribution of Lebesgue Probability Measures on Compact Leaves of the Horocycle Foliations of Bianchi 3-Orbifolds

  • Otto Romero
  • Alberto VerjovskyEmail author
Article
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Abstract

Inspired by the works of Zagier (Eisenstein series and the Riemann zeta function, Automorphic forms, representation theory and arithmetic, Tata Institute of Fundamental Research, Bombay 1979, Springer, New York, 1981) and Sarnak (Commun Pure Appl Math, 34:719–739, 1981), we study the probability measures \(\nu (t)\) with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle \(\mathcal {F}({M}_D)\) of 3-dimensional hyperbolic Bianchi orbifolds \({M}_D=\mathbb {H}^3/\widetilde{\Gamma }_D\), of finite volume and with only one cusp. Here \(\widetilde{\Gamma }_D\subset \mathbf {PSL}(2,\mathbb {C})\) is the Bianchi group corresponding to the imaginary quadratic field \(\mathbb {Q}(\sqrt{-D})\). Thus \(\widetilde{\Gamma }_D\) consist of Möbius transformations with coefficients in the ring of integers of \(\mathbb {Q}(\sqrt{-D})\). If \(l \in \mathbb {N}\), \(k,m \in \mathbb {Z}\) are such that \(k,m \in [-l,l]\), the appropriate Eisenstein series \(\widehat{E}_{km}^l ( g,s)\) (which are defined and analytic for \(\mathrm {Re}(s) \ge 1\)) admit an analytic continuation to all of \(\mathbb {C}\), except when \(l=k=m=0\) in which case there is a pole for \(s=1\). Using this fact we show that the elliptic curves which are the compact orbits of the complex horocycle flow \(h_T:\mathcal {F}({M}_D)\longrightarrow \mathcal {F}({M}_D)\) (\(T \in \mathbb {C}\)) are expanded by the real geodesic flow \(g_t, t\in \mathbb {R}\), and they become equidistributed in \(\mathcal {F}({M}_D)\) with respect to the normalized Haar measure as \(t \longrightarrow \infty \). This follows from the equidistribution of Lebesgue probability measures on compact leaves of the horocycle foliations in the orthonormal frame bundle of \(M_D\), which is equal to the quotient \(\mathbf {PSL}(2,\mathbb {C})/\widetilde{\Gamma }_D\). The same equidistribution property occurs for the spin bundle of \(M_D\) which is the homogeneous space \(\mathbf {SL}(2,\mathbb {C})/\Gamma _D\), where \(\Gamma _D\) is the Bianchi subgroup in \(\mathbf {SL}(2,\mathbb {C})\) consisting of matrices with entries in the ring of integers of \(\mathbb {Q}(\sqrt{-D})\). Our method uses the theory of spherical harmonics in the unit tangent bundle orbifold \(T_1(M_D)={\mathbf {SO}}(2){\backslash }\mathbf {PSL}(2,\mathbb {C})/\widetilde{\Gamma }_D=T_1(\mathbb {H}^3)/{\Gamma ^*_D}\), where \({\Gamma ^*_D}\) is the action of \({\widetilde{\Gamma }}_D\) on the unit tangent bundle of \(\mathbb {H}^3\) via the differential of the elements of \(\widetilde{\Gamma }_D\).

Keywords

Complex horocycle flows and foliations Bianchi groups and 3-orbifolds Equidistribution 

Mathematics Subject Classification

37D40 51M10 11M36 

Notes

Acknowledgements

This paper contains part of the Ph. D. thesis of the first author. He would like to thank his advisor, the second author Alberto Verjovsky for his suggestion to work on this topic, his constant support, and sharing some of his ideas on this subject. Also for the first author (OR) were very important the seminars and advise of Dr. Santiago López de Medrano, and the help and patience of Dr. Gregor Weingart. He also thanks Dr. Fuensanta Aroca for a fellowship from her PAPIIT grant IN108216. The second author (AV) benefited from a PAPIIT (DGAPA, Universidad Nacional Autónoma de México) grant IN106817. Both authors would like to thank Professor Paul Garrett for his comments and references regarding Eisenstein series, and the anonymous referee for his/her suggestions.

References

  1. Anosov, D.V.: Geodesic flows on closed Riemann manifolds with negative curvature,  Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder American Mathematical Society, Providence, R.I., p.p. iv+235 (1969)Google Scholar
  2. Bianchi, L.: Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari. Math. Ann. 40(3), 332–412 (1892)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bowen, R., Marcus, B.: Unique ergodicity for horocycle foliations. Israel J. Math. 26(1), 43–67 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bruggeman, R., Motohashi, Y.: Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field. Functiones et Approximatio Commentarii Mathematici 31, 23–92 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dal’Bo F.: Geodesic and Horocyclic Trajectories, Translated from the 2007 French original. Universitext. Springer, London, EDP Sciences, Les Ulis, pp. xii+176 (2011)Google Scholar
  6. Dani, S.G.: Dynamical systems on homogeneous spaces. Bull. Am. Math. Soc. 82(6), 950–952 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dani, S.G.: Invariant measures and minimal sets of horospherical flows. Invent. Math. 64(2), 357–385 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dani, S.G., Smillie, J.: Uniform distribution of horocycle orbits for Fuchsian groups. Duke Math. J. 51, 185–194 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Elstroedt, J., Grunewald, F., Mennicke, J.: Groups acting on hyperbolic space: harmonic analysis and number theory. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  10. Estala, S.: Distribution of cusp sections in the Hilbert modular orbifold. J. Number Theory 155, 202–225 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Flaminio, L., Forni, G.: Invariant distributions and time averages for horocycle flows. Duke Math. J. 119(3), 465–526 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Furstenberg, H.: The unique ergodicity of the horocycle flow, Recent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972), Lecture Notes in Math, Vol. 318. Springer, Berlin, pp. 95–115 (1973)Google Scholar
  13. Ghys, É.: Holomorphic Anosov systems. Invent. Math. 119(3), 585–614 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Garrett, P.: Intertwinings among principal series of \({\mathbf{SL}_2(\mathbb{C}})\). http://www-users.math.umn.edu/~Garrett/m/v/
  15. Ghys, É., Verjovsky, A.: Locally free holomorphic actions of the complex affine group, Geometric study of foliations (Tokyo, 1993), World Sci. Publ., River Edge, NJ, pp. 201–217 (1994)Google Scholar
  16. Guleska, L.: Thesis: Sum formula for SL2 over imaginary quadratic number fields, Utrecht (2004)Google Scholar
  17. Langlands, R.: On the functional equations satisfied by Eisenstein series, Lecture Notes in Math, vol. 544. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  18. Maucourant, F., Schapira, B.: On topological and measurable dynamics of unipotent frame flows for hyperbolic manifolds, 2017–2019 (2017). \(<\)hal-01455739\(>\) Google Scholar
  19. Moeglin, C., Waldspurger, J.: Spectral decomposition and eisenstein series. Cambridge Univ. Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  20. Ratner, M.: On Raghunathan’s measure conjecture. Ann. Math. 134(3), 545–607 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Reid, A.W.: Arithmeticity of Knot Complements. J. Lond. Math. Soc. (2) 43(1), 171–184 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Sarnak, P.: Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series. Commun. Pure Appl. Math. 34, 719–739 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Stark, H.M.: A historical note on complex quadratic fields with class-number one. Proc. Am. Math. Soc. 21, 254–255 (1969)MathSciNetzbMATHGoogle Scholar
  25. Stark, H.M.: A complete determination of the complex quadratic fields of class-number one. Michigan Math. J. 14, 1–27 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Starkov, A.N.: Dynamical systems on homogeneous spaces, translated from the 1999 Russian original by the author. Translations of Mathematical Monographs (Book 190). American Mathematical Society, Providence, RI, pp. xvi+243 (2000)Google Scholar
  27. Thurston, W.P.: The geometry and topology of 3-manifolds, lecture notes. Princeton University, Princeton (1978)Google Scholar
  28. Wigner, E.: Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, Cambridge (1959)zbMATHGoogle Scholar
  29. Zagier, D.: Eisenstein series and the Riemann zeta function, automorphic forms, representation theory and arithmetic, Tata Institute of Fundamental Research, Bombay 1979, pp. 275–301. Springer, New York (1981)Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticasUnidad Cuernavaca, UNAMCuernavacaMexico

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