Israel Journal of Mathematics

, Volume 114, Issue 1, pp 205–220

On the concept of level for subgroups of SL2 over arithmetic rings

Article

Abstract

We define the concept of level for arbitrary subgroups Γ of finite index in the special linear group SL2(AS), whereAS is the ring ofS-integers of a global fieldk provided thatk is an algebraic number field, or card (S) ≥ 2. It is shown that this concept agrees with the usual notion of ‘Stufe’ for congruence subgroups. In the case SL2(O),O the ring of integers of an imaginary quadratic number field, this criterion for deciding whether or not an arbitrary subgroup of finite index is a congruence subgroup is used to determine the minimum of the indices of non-congruence subgroups.

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References

  1. [1]
    H. Bass,K-theory and stable algebra, Publications Mathématiques de l’Institut des Hautes Études Scientifiques22 (1964), 5–60.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    R. Fricke,Über die Substitutionsgruppen, welche zu den aus dem Legendreschen Integralmodul k 2(w)gezogenen Wurzeln gehören, Mathematische Annalen28 (1887), 99–118.CrossRefGoogle Scholar
  3. [3]
    F. Grunewald and J. Schwermer,Free non-abelian quotients of SL2 over orders of imaginary quadratic number fields, Journal of Algebra69 (1981), 298–304.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    F. Grunewald and J. Schwermer,Arithmetic quotients of hyperbolic 3-space, cusp forms and link complements, Duke Mathematical Journal48 (1981), 351–358.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    F. Grunewald and J. Schwermer,Subgroups of Bianchi groups and arithmetic quotients of hyperbolic 3-space, Transactions of the American Mathematical Society335 (1993), 47–78.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    B. Huppert,Endliche Gruppen I, Grundlehren der Mathematischen Wissenschaften, Bd. 134, Springer, Berlin-Heidelberg-New York, 1967.MATHGoogle Scholar
  7. [7]
    A. Hurwitz,Die unimodularen Substitutionen in einem algebraischen Zahlkörper, Nachrichten von der k. Gesellschaft der Wissenschaften zu Göttingen Mathematische Physikalische Klasse (1895), 332–356 (= Math. Werke, Bd. II, Basel (1933), 244–268).Google Scholar
  8. [8]
    F. Klein and R. Fricke,Vorlesungen über die Theorie der elliptischen Modulfunktionen, Bd. I, Leipzig, 1890.Google Scholar
  9. [9]
    A. Lubotzky,Free quotients and the congruence kernel of SL2, Journal of Algebra77 (1982), 411–418.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    A. W. Mason,Congruence hulls in SLn, Journal of Pure and Applied Algebra89 (1993), 255–272.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Newman,Maximal normal subgroups of the modular group, Proceedings of the American Mathematical Society19 (1968), 1138–1144.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    M. Newman,Integral Matrices, Academic Press, New York-London, 1972.MATHGoogle Scholar
  13. [13]
    R. Phillips and P. Sarnack,Spectrum of Fermat curves, Geometric and Functional Analysis1 (1991), 79–146.CrossRefGoogle Scholar
  14. [14]
    G. Pick,Über gewisse ganzzahlige lineare Substitutionen, welche sich nicht durch algebraische Congruenzen erklären lassen, Mathematische Annalen28 (1887), 119–124.CrossRefMathSciNetGoogle Scholar
  15. [15]
    R. A. Rankin,Lattice subgroups of free congruence subgroups, Inventiones Mathematicae2 (1967), 215–221.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    R. Riley,Applications of a computer implementation of Poincaré’s polyhedron, Mathematics of Computation40 (1983), 607–632.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. Rosenlicht,Equivalence relations on algebraic curves, Annals of Mathematics56 (1962), 169–191.CrossRefMathSciNetGoogle Scholar
  18. [18]
    J-P. Serre,Le problème des groupes de congruence pour SL2, Annals of Mathematics92 (1970), 489–527.CrossRefMathSciNetGoogle Scholar
  19. [19]
    R. G. Swan,Generators and relations for certain special linear groups, Advances in Mathematics6 (1971), 1–77.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    K. Vogtmann,Rational homology of Bianchi groups, Mathematische Annalen272 (1985), 399–419.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    K. Wohlfahrt,Über Dedekindsche Summen und Untergruppen der Modulgruppe, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg23 (1959), 5–10.MATHMathSciNetGoogle Scholar
  22. [22]
    K. Wohlfahrt,An extension of F. Klein’s level concept, Illinois Journal of Mathematics8 (1964), 529–535.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1999

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität DüsseldorfDüsseldorfGermany
  2. 2.Mathematisches InstitutUniversität DüsseldorfDüsseldorfGermany

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