The topological trace formula. (English) Zbl 1097.14017

Goresky, Mark; MacPherson, Robert

J. Reine Angew. Math. 560, 77-150 (2003).

Summary: The topological trace formula is a computation of the Lefschetz number of a Hecke correspondence \(C\) acting on the weighted cohomology groups, defined in M. Goresky, G. Harder and R. MacPherson [Invent. Math. 116, No. 1–3, 139–213 (1994; Zbl 0849.11047)], of a locally symmetric space \(X\). It expresses this Lefschetz number as a sum of contributions from fixed-point components of \(C\) on the reductive Borel Serre compactification of \(X\). The proof uses the Lefschetz fixed-point formula of M. Goresky and R. MacPherson [Invent. Math. 111, No. 1, 1–33 (1993; Zbl 0786.55001)].

MSC:

14F43	Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
11F75	Cohomology of arithmetic groups
22E40	Discrete subgroups of Lie groups
32S60	Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)

Citations:

Zbl 0849.11047; Zbl 0786.55001

References:

[1]	Ar J., Inv. Math. 97 pp 257– (1989)
[2]	Ar J, Duke Math. J. 45 pp 911– (1978)
[3]	[Ar3] J. Arthur, Characters, harmonic analysis, and an L2-Lefschetz formula, The Mathematical Heritage of Hermann Weyl, Proc. Symp. Pure Math.48, Amer. Math. Soc., Providence, RI, 1998.
[4]	[AMRT] A. Ash, D. Mumford, M. Rapoport, and Y.S. Tai, Smooth Compactification of Locally Symmetric Varieties, Math. Sci. Press, Brookline, Mass., 1975. · Zbl 0334.14007
[5]	Ba D., J. Funct. Anal. 53 pp 151– (1983)
[6]	Be J., Bonner Math. Schr. 164 pp 1985–
[7]	[Bo3] A. Borel, Linear Algebraic Groups (second edition), Grad. Texts Math.126, Springer Verlag, New York 1991.
[8]	Bo A, Norm. Sup. 7 pp 235– (1974)
[9]	[Bo5] A. Borel, Sheaf theoretic intersection cohomology, Seminar on Intersection Cohomology, Progr. Math. 50, Birkh user, Boston 1984.
[10]	Borel J.-P., Comm. Math. Helv. 48 pp 436– (1973)
[11]	Indag. Math. pp 399– (1958)
[12]	Norm. Sup. 31 pp 181– (1998)
[13]	[GeM] S. Gelfand and Y. Manin, Homological Algebra, in: Algebra V, Encycl. Math. Sci. 38, Springer-Verlag, Berlin 1994.
[14]	[Gi] C. Gibson, K. Wirthm ller, A. Du Plessis, E. Looijenga, Topological Stability of Smooth Mappings, Lect. Notes Math.552, Springer Verlag, New York 1976.
[15]	Geoghegan A., Amer. J. Math 116 pp 397– (1994)
[16]	Goresky G., Inv. Math. 116 pp 139– (1994)
[17]	Goresky R., Duke Math. J. 89 pp 477– (1997)
[18]	[GM1] M. Goresky and R. MacPherson, Lefschetz numbers of Hecke correspondences, in: The Zeta Functions of Picard Modular Surfaces, R. Langlands and D. Ramakrishnan, eds., Publ. C.R.M., Univ. de Montreal (1992), 465-478. · Zbl 0828.14029
[19]	Goresky R., Inv. Math. 111 pp 1– (1993)
[20]	[GM3] M. Goresky and R. MacPherson, Stratified Morse Theory, Ergeb. Math.14, Springer Verlag, Heidelberg 1989.
[21]	Goresky R., Inv. Math. 71 pp 77– (1983)
[22]	[GM5] M. Goresky and R. MacPherson, Errata to: Local contribution to the Lefschetz fixed point formula, 2003, http://arxiv.org/math.AT.0301289.
[23]	[GI] A. Grothendieck, L. Illusie, Formule de Lefschetz, S.G.A. 5, Cohomologie l-adique et Fonctions L, Lect. Notes Math. 589, Springer Verlag, Heidelberg (1977), 73-137.
[24]	[H] G. Harder, Anhang (Letter to Goresky and MacPherson) inEisensteinkohomologie und die Konstruktion gemischter Motive, Lect. Notes Math. 1562, Springer Verlag, Heidelberg 1993.
[25]	[I] B. Iverson, Cohomology of Sheaves, Universitext, Springer-Verlag, Heidelberg1986.
[26]	[KS] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundl. Math. Wiss.292, Springer Verlag, Heidelberg 1990.
[27]	[KN] S. Kobayashi and K. Nomizu, Foundations of Di erential Geometry I, II, John Wiley and Sons, New York1966.
[28]	Ko B, Ann. Math. 74 pp 329– (1961)
[29]	Ku M., Amer. J. Math 94 pp 486– (1972)
[30]	[L1] R. Langlands, Modular forms and l-adic representations, in: Modular Functions of One Variable II, Lect. Notes Math. 349, Springer Verlag, New York 1973.
[31]	[L2] R. Langlands, On the functional equation satisfied by Eisenstein series, Lect. Notes Math. 544, Springer Verlag, New York 1976.
[32]	Le E, Inv. Math. 121 pp 389– (1995)
[33]	Le E, Enseign. Math. 42 pp 201– (1996)
[34]	Lo E., Comp. Math. 67 pp 3– (1988)
[35]	Mac R, Ann. Math. 100 pp 423– (1974)
[36]	[Mat] J. Mather, Notes on Topological Stability, mimeographed notes, Harvard University, 1970.
[37]	[Mo] H. Moscovici, Lefschetz formula for Hecke operators, Lie Groups and Representations III, Lect. Notes Math. 1077, Springer Verlag, New York (1984), 321-258.
[38]	Ann. Math. 150 pp 1– (1999)
[39]	No K, Ann. Math. 59 pp 531– (1954)
[40]	No K, Amer. J. Math. 76 pp 33– (1954)
[41]	[R] M. Rapoport, On the shape of the contribution of a fixed point on the boundary: the case of Q-rank one, in: The Zeta Functions of Picard Modular Surfaces, R. Langlands and D. Ramakrishnan, eds., Publ. C. R. M., Univ. de Montreal (1992), 465-478.
[42]	Sa L, Comm. Math. Helv. 72 pp 167– (1997)
[43]	Sa L., Ann. Math 132 pp 1– (1990)
[44]	Sat I, Ann. Math. 72 pp 555– (1960)
[45]	[Sh] S. Shokranian, Lefschetz numbers on weighted cohomology for twisted groups, preprint, Dept. Matematica, Univ. de Brasilia, 1999.
[46]	[Sp] T. A. Springer, Linear Algebraic Groups, Progr. Math.9, Birkh user, Boston, MA, 1981.
[47]	St M, Inv. Math. 115 pp 241– (1994)
[48]	[Ve] J. L. Verdier, The Lefschetz fixed point formula in etale cohomology, Proceedings of Conference on Local Fields, T. A. Springer, ed., Lect. Notes Math. 192, Springer Verlag, Heidelberg (1971), 165-168.
[49]	[Vo] D. Vogan, Representations of Real Reductive Lie Groups, Progr. Math.15, Birkh user, Boston, MA, 1981. · Zbl 0469.22012
[50]	[Wa] G. Warner, Harmonic Analysis on Semisimple Lie Groups I, Grundl. Math. Wiss.188, Springer Verlag, Heidelberg 1972.
[51]	[We] U. Weselmann, thesis, University of Bonn, 1996, http://www.mathi.uni-heidelberg.de/ weselman/.
[52]	Zucker, Inv. Math. 70 pp 169– (1982)
[53]	Zucker, Comp. Math. 59 pp 339– (1986)
[54]	Zucker, Comm. Math. Helv. 58 pp 312– (1983)