Betti numbers of congruence groups. (Appendix: On representations of compact \(p\)-adic groups by Ze’ev Rudnick).

*(English)*Zbl 0843.11027In general very little is known on automorphic multiplicities. Up to now there are mainly asymptotic assertions on varying the representation or the group. This remarkable paper contains a new point of view. Varying the group it is given an assertion on the shape of the function (group \(\mapsto\) automorphic multiplicity).

More precisely let \(G\) be a real or \(p\)-adic semisimple Lie group and \(\Gamma \subset G\) a cocompact lattice. The right regular representation of \(G\) on \(L^2 (\Gamma \setminus G)\) decomposes into irreducibles with finite multiplicities \(m_\Gamma (\pi)\). A representation \(\pi\) is called cohomological if its isotype in \(L^2 (\Gamma \setminus G)\) contributes to the cohomology of \(\Gamma\). Let \(\Gamma (N)\) be a family of congruence subgroups and \(H_N\) the finite group \(\Gamma/ \Gamma (N)\). Then \(H_N\) acts on the isotypes \(L^2 (\Gamma (N) \setminus G) (\pi)\) and has finite multiplicities \(m_{H_N} (\rho, \pi)\), \(\rho\in \widehat {H}_N\) there. For \(d\geq 1\) define the \(d\)-dimensional part of \(m_\Gamma (\pi)\) by \[ {}^d m_{\Gamma (N)} (\pi):= \sum _{\substack{ \rho\in \widehat {H}_N\\ \dim\rho =d }} m_{H_N} (\rho, \pi). \] Now call a sequence \(a_n\) polynomial periodic sequences if there are periodic \(b_j (n)\), \(j= 1, \dots, q\) such that \(a_n= \sum^q_{j=0} b_j (n) n^j\). The main theorem of the paper asserts that if \(G\) is \(p\)-adic then for fixed \(d\) and \(\pi\) the sequence \(N\mapsto^d m_{\Gamma (N)} (\pi)\) is polynomial periodic. The same holds for real \(G\) if \(\pi\) is cohomological. The proof uses a result of A. Lubotsky and A. Magid [Varieties of representations of finitely generated groups, Mem. Am. Math. Soc. 336 (1985; Zbl 0598.14042)] stating that all irreducible representations of \(\Gamma\) of dimension \(d\) factor through a fixed quotient \(\Delta\) which is abelian by finite. The representations of \(\Delta\) can be parametrized by the tori of characters of finite index subgroups. The representations with finite image correspond to torsion points in those tori. The representations corresponding to an isotype \(\pi\) make up algebraic sets in these tori and the torsion points in them can be found in a union of subgroups. This finally gives the claim.

More precisely let \(G\) be a real or \(p\)-adic semisimple Lie group and \(\Gamma \subset G\) a cocompact lattice. The right regular representation of \(G\) on \(L^2 (\Gamma \setminus G)\) decomposes into irreducibles with finite multiplicities \(m_\Gamma (\pi)\). A representation \(\pi\) is called cohomological if its isotype in \(L^2 (\Gamma \setminus G)\) contributes to the cohomology of \(\Gamma\). Let \(\Gamma (N)\) be a family of congruence subgroups and \(H_N\) the finite group \(\Gamma/ \Gamma (N)\). Then \(H_N\) acts on the isotypes \(L^2 (\Gamma (N) \setminus G) (\pi)\) and has finite multiplicities \(m_{H_N} (\rho, \pi)\), \(\rho\in \widehat {H}_N\) there. For \(d\geq 1\) define the \(d\)-dimensional part of \(m_\Gamma (\pi)\) by \[ {}^d m_{\Gamma (N)} (\pi):= \sum _{\substack{ \rho\in \widehat {H}_N\\ \dim\rho =d }} m_{H_N} (\rho, \pi). \] Now call a sequence \(a_n\) polynomial periodic sequences if there are periodic \(b_j (n)\), \(j= 1, \dots, q\) such that \(a_n= \sum^q_{j=0} b_j (n) n^j\). The main theorem of the paper asserts that if \(G\) is \(p\)-adic then for fixed \(d\) and \(\pi\) the sequence \(N\mapsto^d m_{\Gamma (N)} (\pi)\) is polynomial periodic. The same holds for real \(G\) if \(\pi\) is cohomological. The proof uses a result of A. Lubotsky and A. Magid [Varieties of representations of finitely generated groups, Mem. Am. Math. Soc. 336 (1985; Zbl 0598.14042)] stating that all irreducible representations of \(\Gamma\) of dimension \(d\) factor through a fixed quotient \(\Delta\) which is abelian by finite. The representations of \(\Delta\) can be parametrized by the tori of characters of finite index subgroups. The representations with finite image correspond to torsion points in those tori. The representations corresponding to an isotype \(\pi\) make up algebraic sets in these tori and the torsion points in them can be found in a union of subgroups. This finally gives the claim.

Reviewer: A.Deitmar (Heidelberg)

##### MSC:

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

22E50 | Representations of Lie and linear algebraic groups over local fields |

##### Keywords:

polynomial periodic sequences; automorphic multiplicities; semisimple Lie group; cocompact lattice; right regular representation; congruence subgroups; tori of characters; torsion points; isotype
PDF
BibTeX
Cite

\textit{P. Sarnak} and \textit{S. Adams}, Isr. J. Math. 88, No. 1--3, 31--72 (1994; Zbl 0843.11027)

Full Text:
DOI

##### References:

[1] | [Ad] S. Adams,Representation varieties of arithmetic groups and polynomial periodicity of Betti numbers, Israel Journal of Mathematics, this issue, pp. 73–124. · Zbl 0858.22016 |

[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. · Zbl 0980.22015 |

[3] | [C-R] C. Curtis and I. Reiner,Methods of Representation Theory, Wiley, New York, 1981. · Zbl 0469.20001 |

[4] | [F-T] R. H. Fox and A. Torres,Dual presentations of the group of a knot, Annals of Mathematics89 (1954), 211–218. · Zbl 0055.16805 |

[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. · Zbl 1246.57002 |

[6] | [F-2] R. H. Fox,Free differential calculus I, II, III, Annals of Mathematics364 (1956), 407–447. · Zbl 0073.25401 |

[7] | [G-G-P] I. Gelfand, M. Graev and I. Piatetsky-Shapiro,Representation Theory and Automorphic Functions, W.B. Saunders, London, 1969. |

[8] | [G] L. Georite,Die Bettischen Zahlen der Zyklische Voerlangerung der Knotenauschenraume, American Journal of Mathematics 56 (1934), 194–198. · Zbl 0009.03902 |

[9] | [Ha] E. Hironaka,Ph.D. Thesis, Brown, 1989. |

[10] | [Ha2] E. Hironaka,Polynomial periodicity for Betti numbers of covering surfaces, Inventiones Mathematicae108 (1992), 289–321. · Zbl 0776.14002 |

[11] | [Ha3] E. Hironaka,Intersection theory on branched covering surfaces and polynomial periodicity, preprint. |

[12] | [Hh] F. Hirzebruch,Arrangements of lines and algebraic surfaces, in Arithmetic Geometry, Vol. II (M. Artin and J. Task, eds.), Birkhäuser, Boston, 1983. · Zbl 0527.14033 |

[13] | [I] M. N. Ishida,The irregularities of Hirzebruch’s surfaces of general type with C 1 2 =3C 2, Mathematische Annalen262 (1983), 407–420. · Zbl 0504.14029 |

[14] | [Jac] N. Jacobson,Basic Algebra II, W. H. Freeman and Co., San Francisco, 1980. |

[15] | [K] M. Kneser,Strong approximation, inAlgebraic Groups, Proceedings of Symposia in Pure MathematicsIX (1966), 187-196. |

[16] | [L1] S. Lang,Introduction to Algebraic and Abelian Functions, Springer-Verlag, Berlin, 1982. |

[17] | [L2] S. Lang,Fundamentals of Diophantine geometry, Springer-Verlag, Berlin, 1983. · Zbl 0528.14013 |

[18] | [Lt] M. Laurant,Equations diophantine exponentielles, Inventiones Mathematicae78 (1984), 299–327. · Zbl 0554.10009 |

[19] | [Lr] A. Libgober,Betti numbers of Abelian covers, Preprint, 1989. |

[20] | [Lr2] A. Libgober,Alexander polynomials of algebraic curves, Duke Mathematical Journal49 (1982), 833–851. · Zbl 0524.14026 |

[21] | [L-M] A. Lubotzky and A. Magid,Varieties of representations of finitely generated groups, Memoirs of the American Mathematical Society58 (1985). · Zbl 0598.14042 |

[22] | [M-M] J. Mayberry and K. Murasugi,Torsion groups of Abelian coverings of links, Transactions of the American Mathematical Society271 (1982), 143–173. · Zbl 0487.57001 |

[23] | [P-s] R. Phillips and P. Sarnak,The spectrum of fermat groups, Geometry Functional Annalysis1 (1991), 80–146. · Zbl 0751.11030 |

[24] | [R-S] D. Ray and I. Singer,R-torsion and the Laplacian, Advances in Mathematics7 (1975), 145–210. · Zbl 0239.58014 |

[25] | [R] Z. Rudnick,Representation varieties of solvable groups, Journal of Pure and Applied Algebra45 (1987), 261–272. · Zbl 0704.20037 |

[26] | [Ru] W. Ruppert,Solving algebraic equations in roots of unity, Journal für die Reine und Angewandte Mathematik435 (1992), 119–156. · Zbl 0763.14008 |

[27] | [Sa] P. Sarnak,Betti numbers of congruence groups, preprint. |

[28] | [S] B. Schoenberg,Elliptic Modular Fanctions, Springer-Verlag, New York, 1974. |

[29] | [Su] D. Sumners,On the homology of cyclic coverings of higher dimensional links, Proceedings of the American Mathematical Society46 (1974), 143–149. · Zbl 0296.57004 |

[30] | [Z] O. Zariski,On the topology of algebraic singularities, American Journal of Mathematics54 (1932), 453–465. · JFM 58.0614.02 |

[31] | [CR] C. Curtis and I. Reiner,Representation Theory of Finite Group and Associative Algebras, Wiley-Interscience, New York, 1962. · Zbl 0131.25601 |

[32] | [H] R. Howe,Kirilov theory for compact p-adic groups, Pacific Journal of Mathematics73 (1977), 365–381. · Zbl 0385.22007 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.