Israel Journal of Mathematics

, Volume 88, Issue 1–3, pp 31–72 | Cite as

Betti numbers of congruence groups



It is shown that, for certain congruence families of Galois coverings of a manifold, the individual Betti numbers are polynomial periodic functions of the level. Similar results are proved for the dimensions of other spaces of automorphic forms.


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  1. [Ad] S. Adams,Representation varieties of arithmetic groups and polynomial periodicity of Betti numbers, Israel Journal of Mathematics, this issue, pp. 73–124.Google Scholar
  2. [B-W] A. Borel and N. Wallach,Continuous cohomology, discrete groups and representations of reductive groups, Annals of Math. Studies 94, Princeton University Press, 1976.Google Scholar
  3. [C-R] C. Curtis and I. Reiner,Methods of Representation Theory, Wiley, New York, 1981.MATHGoogle Scholar
  4. [F-T] R. H. Fox and A. Torres,Dual presentations of the group of a knot, Annals of Mathematics89 (1954), 211–218.MathSciNetGoogle Scholar
  5. [F-1] R. H. Fox,A quick trip through knot theory, in Topology of 3-Manifolds (M. K. Fort, Jr., ed.), Prentice-Hall, Englewood Cliffs, NJ, 1961.Google Scholar
  6. [F-2] R. H. Fox,Free differential calculus I, II, III, Annals of Mathematics364 (1956), 407–447.CrossRefGoogle Scholar
  7. [G-G-P] I. Gelfand, M. Graev and I. Piatetsky-Shapiro,Representation Theory and Automorphic Functions, W.B. Saunders, London, 1969.Google Scholar
  8. [G] L. Georite,Die Bettischen Zahlen der Zyklische Voerlangerung der Knotenauschenraume, American Journal of Mathematics 56 (1934), 194–198.CrossRefMathSciNetGoogle Scholar
  9. [Ha] E. Hironaka,Ph.D. Thesis, Brown, 1989.Google Scholar
  10. [Ha2] E. Hironaka,Polynomial periodicity for Betti numbers of covering surfaces, Inventiones Mathematicae108 (1992), 289–321.MATHCrossRefMathSciNetGoogle Scholar
  11. [Ha3] E. Hironaka,Intersection theory on branched covering surfaces and polynomial periodicity, preprint.Google Scholar
  12. [Hh] F. Hirzebruch,Arrangements of lines and algebraic surfaces, in Arithmetic Geometry, Vol. II (M. Artin and J. Task, eds.), Birkhäuser, Boston, 1983.Google Scholar
  13. [I] M. N. Ishida,The irregularities of Hirzebruch's surfaces of general type with C 12=3C 2, Mathematische Annalen262 (1983), 407–420.MATHCrossRefMathSciNetGoogle Scholar
  14. [Jac] N. Jacobson,Basic Algebra II, W. H. Freeman and Co., San Francisco, 1980.MATHGoogle Scholar
  15. [K] M. Kneser,Strong approximation, inAlgebraic Groups, Proceedings of Symposia in Pure MathematicsIX (1966), 187-196.Google Scholar
  16. [L1] S. Lang,Introduction to Algebraic and Abelian Functions, Springer-Verlag, Berlin, 1982.MATHGoogle Scholar
  17. [L2] S. Lang,Fundamentals of Diophantine geometry, Springer-Verlag, Berlin, 1983.MATHGoogle Scholar
  18. [Lt] M. Laurant,Equations diophantine exponentielles, Inventiones Mathematicae78 (1984), 299–327.CrossRefMathSciNetGoogle Scholar
  19. [Lr] A. Libgober,Betti numbers of Abelian covers, Preprint, 1989.Google Scholar
  20. [Lr2] A. Libgober,Alexander polynomials of algebraic curves, Duke Mathematical Journal49 (1982), 833–851.MATHCrossRefMathSciNetGoogle Scholar
  21. [L-M] A. Lubotzky and A. Magid,Varieties of representations of finitely generated groups, Memoirs of the American Mathematical Society58 (1985).Google Scholar
  22. [M-M] J. Mayberry and K. Murasugi,Torsion groups of Abelian coverings of links, Transactions of the American Mathematical Society271 (1982), 143–173.MATHCrossRefMathSciNetGoogle Scholar
  23. [P-s] R. Phillips and P. Sarnak,The spectrum of fermat groups, Geometry Functional Annalysis1 (1991), 80–146.MATHCrossRefMathSciNetGoogle Scholar
  24. [R-S] D. Ray and I. Singer,R-torsion and the Laplacian, Advances in Mathematics7 (1975), 145–210.CrossRefMathSciNetGoogle Scholar
  25. [R] Z. Rudnick,Representation varieties of solvable groups, Journal of Pure and Applied Algebra45 (1987), 261–272.MATHCrossRefMathSciNetGoogle Scholar
  26. [Ru] W. Ruppert,Solving algebraic equations in roots of unity, Journal für die Reine und Angewandte Mathematik435 (1992), 119–156.MathSciNetGoogle Scholar
  27. [Sa] P. Sarnak,Betti numbers of congruence groups, preprint.Google Scholar
  28. [S] B. Schoenberg,Elliptic Modular Fanctions, Springer-Verlag, New York, 1974.Google Scholar
  29. [Su] D. Sumners,On the homology of cyclic coverings of higher dimensional links, Proceedings of the American Mathematical Society46 (1974), 143–149.MATHCrossRefMathSciNetGoogle Scholar
  30. [Z] O. Zariski,On the topology of algebraic singularities, American Journal of Mathematics54 (1932), 453–465.CrossRefMathSciNetGoogle Scholar
  31. [CR] C. Curtis and I. Reiner,Representation Theory of Finite Group and Associative Algebras, Wiley-Interscience, New York, 1962.Google Scholar
  32. [H] R. Howe,Kirilov theory for compact p-adic groups, Pacific Journal of Mathematics73 (1977), 365–381.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1994

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of MinnesotaMinneapolisUSA

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