zbMATH — the first resource for mathematics

The logarithm of the Dedekind \(\eta\)-function. (English) Zbl 0648.58035
The appearance of Dedekind sums in topology was first noted by F. Hirzebruch [Prospects Math., Ann. Math. Stud. 70, 3-31 (1971; Zbl 0252.58009)]. These sums were introduced by Dedekind to describe the transformation of \(\log\eta(\tau)\) under elements of \(SL(2,{\mathbb{Z}})\), where \(\eta(\tau)\) is the Dedekind eta-function. In the present paper the appearance of \(\log\eta(\tau)\) in topology is described in the general context of index theory and related to several other invariants. The paper deals with several generalizations of the signature theorem involving the case of manifolds with boundary, the equivariant case and families of elliptic operators. The results are applied to a fibration \(Z\to^{M}X\), where the fibre M is a torus, and X is a compact surface with boundary. The local coefficient system given by \(H^ 1(M)\) then arises from a representation \(\pi_ 1(X)\to SL(2,{\mathbb{Z}})\). The signature of Z is equal to the signature of the local coefficient system over X given by \(H^ 1(M)\), and there is a function \(\Phi\) : SL(2,\({\mathbb{Z}})\to {\mathbb{Q}}\), defined by W. Meyer [Math. Ann. 201, 239-264 (1973; Zbl 0241.55019)] such that \[ sign Z=sign (X,H^ 1(M))=- \sum \Phi (A), \] where the sum is over the monodromy matrices A around the boundary circles S of X by the action on \(H^ 1(M)\). There are several other invariants \(SL(2,{\mathbb{Z}})\to {\mathbb{Q}}\) defined in this context and described in the paper: The invariant \(\eta(A)\) which is the Atiyah-Patodi-Singer spectral invariant of the component W(A) of \(\partial Z\) corresponding to the circle with monodromy matrix A. The invariant \(\eta^ 0(A)\) which arises as an adiabatic limit from a family of \(\eta\)-invariants for W(A) as studied by J.-M. Bismut and D. S. Freed [Commun. Math. Phys. 107, 103-163 (1986)]. The invariant \(\chi(A)\) which describes essentially the transformation properties of \(\log\eta(\tau)\) under A. The signature defect \(\delta(A)\) defined by F. Hirzebruch. The invariant \(\mu(A)\) describing the logarithmic monodromy (divided by \(\pi^ i\)) of Quillen’s determinant bundle, which is a complex line bundle over X associated to the signature operator on Z. The value \(L_ A(0)\) of the Shimiuzu L-function. Some identities between these invariants were proved by different authors. The main result of the present paper is that they all coincide if \(A\in SL(2,{\mathbb{Z}})\) is hyperbolic.
Reviewer: K.H.Mayer

58J20 Index theory and related fixed-point theorems on manifolds
11F11 Holomorphic modular forms of integral weight
Full Text: DOI EuDML
[1] Atiyah, M.F.: The signature of fibre-bundles. Global Analysis, 73-84. Tokyo, Princeton: University Press 1969 · Zbl 0193.52302
[2] Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. II. Ann. Math.88, 451-491 (1968) · Zbl 0167.21703 · doi:10.2307/1970721
[3] Atiyah, M.F., Singer, I.M.: The index of elliptic operators. I, III. Ann. Math.87, 484-530, 546-604 (1968) · Zbl 0164.24001 · doi:10.2307/1970715
[4] Atiyah, M.F., Singer, I.M.: The index of elliptic operators. IV. Ann. Math.93, 119-138 (1971) · Zbl 0212.28603 · doi:10.2307/1970756
[5] Atiyah, M.F., Donnelly, H., Singer, I.M.: Eta invariants, signature defects of cusps and values of L-functions. Ann. Math.118, 131-177 (1983) · Zbl 0531.58048 · doi:10.2307/2006957
[6] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Phil. Soc.77, 43-69 (1975) · Zbl 0297.58008 · doi:10.1017/S0305004100049410
[7] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Phil. Soc.78, 405-432 (1975) · Zbl 0314.58016 · doi:10.1017/S0305004100051872
[8] Bismut, J.M.: The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs. Invent. Math.83, 91-151 (1986) · Zbl 0592.58047 · doi:10.1007/BF01388755
[9] Bismut, J.M., Freed, D.S.: The analysis of elliptic families, I. Metrics and connections on determinant bundles. Commun. Math. Phys.106, 159-176 (1986) · Zbl 0657.58037 · doi:10.1007/BF01210930
[10] Bismut, J.M., Freed, D.S.: The analysis of elliptic families. II. Commun. Math. Phys.107, 103-163 (1986) · Zbl 0657.58038 · doi:10.1007/BF01206955
[11] Chern, S.S., Hirzebruch, F., Serre, J.-P.: The index of a fibered manifold. Proc. Am. Math. Soc.8, 587-596 (1957) · Zbl 0083.17801 · doi:10.1090/S0002-9939-1957-0087943-0
[12] Dedekind, R.: Erläuterungen zu zwei Fragmenten von Riemann. Riemann’s gesammelte Mathematische Werke 2, 466-478. New York: Dover 1982
[13] Deligne, P.: Equations différentielles à points singulier réguliers. Lect. Notes Math. 163. Berlin, Heidelberg, New York: Springer 1970
[14] Donnelly, H., Patodi, V.K.: Spectrum and the fixed point sets of isometries II. Topology16, 1-12 (1977) · Zbl 0341.53023 · doi:10.1016/0040-9383(77)90027-1
[15] Freed, D.S. (to appear)
[16] Donaldson, S.K.: Infinite determinants, stable bundles and curvatur. Duke. Math. J.54, 231-247 (1987) · Zbl 0627.53052 · doi:10.1215/S0012-7094-87-05414-7
[17] Hirzebruch, F.: Hilbert modular surfaces. Enseign. Math.19, 183-281 (1973) · Zbl 0285.14007
[18] Hirzebruch, F.: The signature theorem: reminiscences and recreation. Prospects in Mathematics. Ann. Math. Stud.70, pp. 3-31. Princeton, University Press 1971 · Zbl 0252.58009
[19] Hochschild, G.: Group extensions of Lie groups. II. Ann. Math.54, 537-551 (1951) · Zbl 0045.30802 · doi:10.2307/1969488
[20] Kodaira, K.: A certain type of irregular algebraic surfaces. J. Analyse Math.19, 207-215 (1967) · Zbl 0172.37901 · doi:10.1007/BF02788717
[21] Kodaira, K.: On compact analytic surfaces. Analytic Functions, 121-136 Princeton: Univ. Press (1960) · Zbl 0137.17401
[22] Lusztig, G.: Novikov’s higher signature theorem and families of elliptic operators. J. Differ. Geometry7, 229-256 (1972) · Zbl 0265.57009
[23] Mackey, G.W.: Les ensembles Borélien et les extensions des groupes. J. Math. Pures. Appl.36, 171-178 (1957) · Zbl 0080.02303
[24] Meyer, C.: Über die Bildung von Klasseninvarianten. Abh. Math. Sem. Univ. Hamburg27, 206-230 (1964) · Zbl 0122.05201 · doi:10.1007/BF02993218
[25] Meyer, C.: Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlenkörpern. Berlin 1957 · Zbl 0079.06001
[26] Meyer, W.: Die Signatur von Flächenbündeln. Math. Ann.201, 239-264 (1973) · Zbl 0245.55017 · doi:10.1007/BF01427946
[27] Meyer, W., Sczech, R.: Über eine topologische und Zahlentheoretische Anwendung von Hirzebruchs Spitzenauflösung. Math. Ann.240, 69-96 (1979) · Zbl 0392.14007 · doi:10.1007/BF01428301
[28] Müller, W.: Signature defects of cusps of Hilbert modular varieties and values of L-functions at s=1. Akademie der Wissenschaften der DDR, preprint 1983 · Zbl 0588.10031
[29] Mumford, D.: Abelian quotients of the Teichmüller modular group. J. d’Analyse Math.18, 227-244 (1967) · Zbl 0173.22903 · doi:10.1007/BF02798046
[30] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31-34 (1985) · Zbl 0603.32016 · doi:10.1007/BF01086022
[31] Quillen, D.: Superconnections and the Chern character. Topology24, 89-95 (1985) · Zbl 0569.58030
[32] Rademacher, K.: Zur Theorie der Dedekindschen Summen. Math. Zeit63, 445-463 (1956) · Zbl 0071.04201 · doi:10.1007/BF01187951
[33] Ray, D.B., Singer, I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145-210 (1971) · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4
[34] Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math.98, 154-177 (1973) · Zbl 0267.32014 · doi:10.2307/1970909
[35] Siegel, C.L.: Advanced Analytic Number Theory. Tata Institute of Fundamental Research, Bombay 1980 · Zbl 0478.10001
[36] Witten, E.: Global gravitational anomalies. Commun. Math. Phys.100, 197-229 (1985) · Zbl 0581.58038 · doi:10.1007/BF01212448
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.