Mathematische Zeitschrift

, Volume 281, Issue 1–2, pp 523–569 | Cite as

Small isospectral and nonisometric orbifolds of dimension 2 and 3

Article

Abstract

Revisiting a construction due to Vignéras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric.

References

  1. 1.
    Agol, I.: Finiteness of arithmetic Kleinian reflection groups. In: Proceedings of the ICM, Madrid, Spain, pp. 951–960 (2006)Google Scholar
  2. 2.
    Agol, I., Culler, M., Shalen, P.B.: Dehn surgery, homology and hyperbolic volume. Algebr. Geom. Topol. 6, 2297–2312 (2006)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Armitage, J.V., Fröhlich, A.: Classnumbers and unit signatures. Mathematika 14, 94–98 (1967)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Belabas, K.: A fast algorithm to compute cubic fields. Math. Comput. Math. Soc. 66(219), 1213–1237 (1997)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Belolipetsky, M., Linowitz, B.: On fields of definition of arithmetic Kleinian reflection groups. II. Int. Math. Res. Not. (2012). doi:10.1093/imrn/rns292
  6. 6.
    Bérard, P.: Transplantation et isospectralité. I. Math. Ann. 292(3), 547–559 (1992)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Borel, A.: Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(1), 1–33 (1981)MATHMathSciNetGoogle Scholar
  8. 8.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bosma, W., de Smit, B.: On arithmeically equivalent number fields of small degree, Algorithmic number theory (Sydney, 2002). In: Lecture Notes in Comput. Sci., vol. 2369, pp. 67–79. Springer, Berlin (2002)Google Scholar
  10. 10.
    Brooks, R., Tse, R.: Isospectral surfaces of small genus. Nagoya Math. J. 107, 13–24 (1987)MATHMathSciNetGoogle Scholar
  11. 11.
    Brueggeman, S., Doud, D.: Local corrections of discriminant bounds and small degree extensions of quadratic base fields. Int. J. Number Theory 4(3), 349–361 (2008)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Brueggeman, S., Doud, D.: Supplementary Tables and Computer Code. http://math.byu.edu/~doud/DiscBound.html
  13. 13.
    Buser, Peter: Geometry and Spectra of Compact Riemann Surfaces. Modern Birkhäuser Classics, Birkhäuser, Boston (2010) (reprint)Google Scholar
  14. 14.
    Buser, P., Flach, N., Semmler, K.-D.: Spectral rigidity and discreteness of 2233-groups, with an appendix by Colin MacLachlan and Gerhard Rosenberger. Math. Proc. Camb. Philos. Soc. 144(1), 145–180 (2008)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Chan, W.K., Xu, F.: On representations of spinor genera. Compos. Math. 140(2), 287–300 (2004)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Chen, S.: Constructing isospectral but nonisometric Riemannian manifolds. Canad. Math. Bull. 35(3), 303–310 (1992)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Chevalley, Claude (1936) L’arithmétique dans les algèbres de matrices. Actual. Sci. Ind. no. 323Google Scholar
  18. 18.
    Chinburg, T., Friedman, E.: The smallest arithmetic hyperbolic three-orbifold. Invent. Math. 86(3), 507–527 (1986)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Chinburg, T., Hamilton, E., Long, D.D., Reid, A.W.: Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds. Duke Math. J. 145, 25–44 (2008)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Chinburg, T., Friedman, E.: An embedding theorem for quaternion algebras. J. Lond. Math. Soc. (2) 60(1), 33–44 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chinburg, T., Friedman, E., Jones, K.N., Reid, A.W.: The arithmetic hyperbolic 3-manifold of smallest volume. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30(1), 1–40 (2001)MATHMathSciNetGoogle Scholar
  22. 22.
    Cohen, H.: A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993)CrossRefGoogle Scholar
  23. 23.
    Cohen, H.: Advanced Topics in Computational Number Theory, Graduate Texts in Math, vol. 193. Springer, New York (2000)CrossRefGoogle Scholar
  24. 24.
    Cohen, H., y Diaz, F.D., Olivier, M.: A table of totally complex number fields of small discriminants. Algorithmic Number Theory. In: 3rd International Symposium, Portland. Lecture Notes in Comp. Sci., vol. 1423, pp. 381–391. Springer, Berlin (1998)Google Scholar
  25. 25.
    Conway, J.H.: The sensual (quadratic) form. Carus Math. Monographs, vol. 26. Math. Assoc. America, Washington, DC (1997)Google Scholar
  26. 26.
    Culler, M., Shalen, P.B.: Singular surfaces, mod 2 homology, and hyperbolic volume, II. Topol. Appl. 158(1), 118–131 (2011)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Culler, M., Shalen, P.B.: 4-free groups and hyperbolic geometry. J. Topol. 5(1), 81–136 (2012)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Darmon, H.: Rational Points on Modular Elliptic Curves, Volume 101 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2004)Google Scholar
  29. 29.
    Deligne, P.: Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, 123–165, Lecture Notes in Math., vol. 244, Springer, Berlin (1971)Google Scholar
  30. 30.
    Deturck, D.M., Gordon, C.S.: Isospectral deformations II: trace formulas, metrics, and potentials, with an appendix by Kyung Bai Lee. Commun. Pure. Appl. Math. 42, 1067–1095 (1989)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Doi, K., Naganuma, H.: On the algebraic curves uniformized by arithmetical automorphic functions. Ann. Math. (2) 86(3), 449–460 (1967)Google Scholar
  32. 32.
    Doyle, P.G., Rossetti, J.P.: Isospectral hyperbolic surfaces have matching geodesics. http://www.math.dartmouth.edu/~doyle/docs/verso/verso
  33. 33.
    Doyle, P.G., Rossetti, J.P.: Laplace-isospectral hyperbolic 2-orbifolds are representation-equivalent. http://www.math.dartmouth.edu/~doyle/docs/rb/rb
  34. 34.
    Dryden, E.B., Strohmaier, A.: Huber’s theorem for hyperbolic orbisurfaces. Can. Math. Bull. 52(1), 66–71 (2009)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Dryden, E.B., Gordon, C.S., Greenwald, S.J., Webb, D.L.: Asymptotic expansion of the heat kernel for orbifolds. Mich. Math. J. 56(1), 205–238 (2008)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29, 39–79 (1975)MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Dunbar, W., Meyerhoff, G.: Volumes of hyperbolic 3-orbifolds. Indiana Univ. Math. J. 43(2), 611–637 (1994)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Ellenberg, J.S., Venkatesh, A.: The number of extensions of a number field with fixed degree and bounded discriminant. Ann. Math. (2) 163(2), 723–741 (2006)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Elstrodt, J., Grunewald, F., Mennicke, J.: Groups Acting on Hyperbolic Space. Springer Monographs in Mathematics. Springer, Berlin (1998) (Harmonic analysis and number theory)Google Scholar
  40. 40.
    Friedman, E.: Analytic formulas for the regulator of a number field. Invent. Math. 98, 599–622 (1989)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Gabai, D., Meyerhoff, R., Milley, P.: Minimum volume cusped hyperbolic three-manifolds. J. Am. Math. Soc. 22(4), 1157–1215 (2009)MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Gabai, D., Meyerhoff, R., Milley, P.: Mom technology and volumes of hyperbolic 3-manifolds. Comment. Math. Helv. 86(1), 145–188 (2011)MATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Gehring, F.W., Martin, G.J.: Minimal co-volume hyperbolic lattices. I. The spherical points of a Kleinian group. Ann. Math. (2) 170(1), 123–161 (2009)MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Gehring, F.W., Maclachlan, C., Martin, G.J., Reid, A.W.: Arithmeticity, discreteness, and volume. Trans. Am. Math. Soc. 349(9), 3611–3643 (1997)MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Giraud, O., Thas, K.: Hearing the shape of drums: physical and mathematical aspects of isospectrality. Rev. Mod. Phys. 82, 2213–2255 (2010)CrossRefGoogle Scholar
  46. 46.
    Carolyn, S.: Gordon, Survey of Isospectral Manifolds, Handbook of Differential Geometry, vol. I. North-Holland, Amsterdam (2000)Google Scholar
  47. 47.
    Gordon, C.S.: Sunada’s isospectrality technique: two decades later. Spectral analysis in geometry and number theory. Contemp. Math. 484. Amer. Math. Soc., Providence, RI, 45–58 (2009)Google Scholar
  48. 48.
    Gordon, C., Mao, Y.: Comparisons of Laplace spectra, length spectra and geodesic flows of some Riemannian manifolds. Math. Res. Lett. 1(6), 677–688 (1994)MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Gordon, C.S., Rossetti, J.P.: Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn’t reveal. Ann. Inst. Fourier Grenoble 53, 2297–2314 (2003)MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Gordon, C., Webb, D.L., Wolpert, S.: One cannot hear the shape of a drum. Bull. Am. Math. Soc. (N.S.) 27(1), 134–138 (1992)MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Gromov, M.: Hyperbolic manifolds according to Thurston and Jørgensen, Seminar Bourbaki 546. In: Lecture Notes in Math, vol. 842. Springer, Berlin (1981)Google Scholar
  52. 52.
    Guo, X., Qin, H.: An embedding theorem for Eichler orders. J. Number Theory 107(2), 207–214 (2004)MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Huber, H.: Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. Math. Ann. 138, 1–26 (1959)MATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Kac, M.: Can one hear the shape of a drum? Am. Math. Mont. 73(4, part II), 1–23 (1966)Google Scholar
  55. 55.
    Katok, S.: Fuchsian Groups. University of Chicago Press, Chicago (1992)MATHGoogle Scholar
  56. 56.
    Kirschmer, M., Voight, J.: Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. (SICOMP) 39(5), 1714–1747 (2010)MATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Klüners, J., Malle, G.: A database for field extensions of the rationals. LMS J. Comput. Math. 4, 182–196 (2001)MATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    Linowitz, B.: Selectivity in quaternion algebras. J. Number Theory 132, 1425–1437 (2012)MATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    LMFDB Collaboration: The l-functions and modular forms database. http://www.lmfdb.org (2013)
  60. 60.
    Louboutin, S.R.: Upper bounds for residues of Dedekind zeta functions and class numbers of cubic and quartic number fields. Math. Comp. 80(275), 1813–1822 (2011)MATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    Maclachlan, C., Rosenberger, G.: Small volume isospectral, nonisometric, hyperbolic 2-orbifolds. Arch. Math. (Basel) 62(1), 33–37 (1994)MATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Maclachlan C., Reid, A.W.: The arithmetic of hyperbolic 3-manifolds. Grad. Texts in Math., vol. 219. Springer, New York (2003)Google Scholar
  63. 63.
    Marshall, T.H., Martin, G.J.: Minimal co-volume hyperbolic lattices. II. Simple torsion in a Kleinian group. Ann. Math. (2) 176(1), 261–301 (2012)MATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Martinet, J.: Petits discriminants des corps de nombres, Number theory days, 1980 (Exeter, 1980), London Math. Soc. In: Lecture Note Ser., vol. 56, pp. 151–193. Cambridge Univ. Press, Cambridge (1980)Google Scholar
  65. 65.
    Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Acad. Sci. U.S.A. 51, 542 (1964)MATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    Mostow, G.D.: Quasi-conformal mappings in \(n\)-space and the rigidity of hyperbolic spaceforms. Publ. I.H.E.S 34, 53–104 (1968)MATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Odlyzko, A.M.: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém. Théor. Nombres Bordx (2) 2(1), 119–141 (1990)MATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    Page, A.: Computing arithmetic Kleinian groups. Math. Comput. 84, 2361–2390 (2015)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Pohst, M.: On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields. J. Number Theory 14(1), 99–117 (1982)MATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    Poitou, G., Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année: (1976, 77), Théorie des nombres, Fasc. 1, Secrétariat Math., Paris (French), pages Exp. No. 6, 18 (1977)Google Scholar
  71. 71.
    Prasad, G.: \(\mathbb{Q}\)-rank one lattices. Invent. Math. 21, 255–286 (1973)MATHMathSciNetCrossRefGoogle Scholar
  72. 72.
    Prasad, G., Rapinchuk, A.S.: Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publ. Math. Inst. Hautes Études Sci. 109, 113–184 (2009)MATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    Rajan, C.S.: On isospectral arithmetical spaces. Am. J. Math. 129(3), 791–808 (2007)MATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Reiner, I.: Maximal Orders. Clarendon Press, Oxford (2003)MATHGoogle Scholar
  75. 75.
    Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)MATHCrossRefGoogle Scholar
  76. 76.
    Shimizu, H.: On zeta functions of quaternion algebras. Ann. Math. (2) 81, 166–193 (1965)MATHCrossRefGoogle Scholar
  77. 77.
    Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math. (2) 85, 58–159 (1967)MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    Sijsling, J.: Arithmetic \((1; e)\)-curves and Belyi maps. Math. Comp. 81(279), 1823–1855 (2012)MATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. Math. (2) 121(1), 169–186 (1985)MATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    Thurston, W.: The geometry and topology of three-manifolds. http://library.msri.org/books/gt3m/ (2002)
  81. 81.
    Takeuchi, K.: Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24(1), 201–212 (1977)MATHMathSciNetGoogle Scholar
  82. 82.
    Vignéras, M.-F.: Variétés riemanniennes isospectrales et non isométriques. Ann. Math. (2) 112(1), 21–32 (1980)MATHCrossRefGoogle Scholar
  83. 83.
    Vignéras, M.-F.: Arithmétique des algèbres de quaternions. In: Lecture Notes in Math, vol. 800. Springer, Berlin (1980)Google Scholar
  84. 84.
    Voight, J.: Enumeration of totally real number fields of bounded root discriminant. Algorithmic number theory. In: Lecture Notes in Comput. Sci., vol. 5011, pp. 268–281. Springer, Berlin (2008)Google Scholar
  85. 85.
    Voight, J.: Totally real number fields. http://www.cems.uvm.edu/~voight/nf-tables/
  86. 86.
    Voight, J.: Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms. In: Quadratic and Higher Degree FormsGoogle Scholar
  87. 87.
    Voight, J.: Computing fundamental domains for Fuchsian groups. J. Théorie Nombres Bordx. 21(2), 467–489 (2009)MathSciNetCrossRefGoogle Scholar
  88. 88.
    Voight, J.: Characterizing quaternion rings over an arbitrary base. J. Reine Angew. Math. 657, 113–134 (2011)MATHMathSciNetGoogle Scholar
  89. 89.
    Voight, J.: Shimura curves of genus at most two. Math. Comp. 78, 1155–1172 (2009)MATHMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA
  3. 3.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA

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