Complexes pondérés sur les compactifications de Baily-Borel: Le cas des variétés de Siegel

Morel, Sophie

doi:10.1090/S0894-0347-06-00538-8

Complexes pondérés sur les compactifications de Baily-Borel: Le cas des variétés de Siegel
HTML articles powered by AMS MathViewer

by Sophie Morel;

J. Amer. Math. Soc. 21 (2008), 23-61

DOI: https://doi.org/10.1090/S0894-0347-06-00538-8

Published electronically: June 9, 2006

PDF | Request permission

Abstract:

In this work, we calculate the trace of a Hecke correspondance composed with a power of the Frobenius endomorphism on the fibre of the intersection complexes of the Baily-Borel compactification of a Siegel modular variety. Our main tool is Pink’s theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncation of the full direct image. More generally, we are able to define analogs in positive characteristic of the weighted cohomology complexes introduced by Goresky, Harder and MacPherson.

References

A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications, Vol. IV, Math Sci Press, Brookline, MA, 1975. MR 457437
W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. MR 216035, DOI 10.2307/1970457
A. A. Beĭlinson, On the derived category of perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 27–41. MR 923133, DOI 10.1007/BFb0078365
Analyse et topologie sur les espaces singuliers. I, Astérisque, vol. 100, Société Mathématique de France, Paris, 1982 (French). MR 751965
J.-L. Brylinski and J.-P. Labesse, Cohomologie d’intersection et fonctions $L$ de certaines variétés de Shimura, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 361–412 (French). MR 777375, DOI 10.24033/asens.1476
Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353, DOI 10.1007/978-3-662-02632-8
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520, DOI 10.1007/BF02684780
Kazuhiro Fujiwara, Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture, Invent. Math. 127 (1997), no. 3, 489–533. MR 1431137, DOI 10.1007/s002220050129
M. Goresky, G. Harder, and R. MacPherson, Weighted cohomology, Invent. Math. 116 (1994), no. 1-3, 139–213. MR 1253192, DOI 10.1007/BF01231560
Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373–444. MR 1124982, DOI 10.1090/S0894-0347-1992-1124982-1
Robert E. Kottwitz and Michael Rapoport, Contribution of the points at the boundary, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 111–150. MR 1155228
R. P. Langlands, Modular forms and $\ell$-adic representations, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin-New York, 1973, pp. 361–500. MR 354617
Marc Levine, Tate motives and the vanishing conjectures for algebraic $K$-theory, Algebraic $K$-theory and algebraic topology (Lake Louise, AB, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 167–188. MR 1367296, DOI 10.1007/978-94-017-0695-7_{7}
Robert P. Langlands and Dinakar Ramakrishnan (eds.), The zeta functions of Picard modular surfaces, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1992. MR 1155223
E. Looijenga and M. Rapoport, Weights in the local cohomology of a Baily-Borel compactification, Complex geometry and Lie theory (Sundance, UT, 1989) Proc. Sympos. Pure Math., vol. 53, Amer. Math. Soc., Providence, RI, 1991, pp. 223–260. MR 1141203, DOI 10.1090/pspum/053/1141203

Complexes d’intersection des compactifications de Baily-Borel : Le cas des groupes unitaires sur $\mathbb {Q}$

Richard Pink, Arithmetical compactification of mixed Shimura varieties, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 209, Universität Bonn, Mathematisches Institut, Bonn, 1990. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1989. MR 1128753
Richard Pink, On $l$-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification, Math. Ann. 292 (1992), no. 2, 197–240. MR 1149032, DOI 10.1007/BF01444618
Richard Pink, On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne, Ann. of Math. (2) 135 (1992), no. 3, 483–525. MR 1166642, DOI 10.2307/2946574
Robert E. Kottwitz and Michael Rapoport, Contribution of the points at the boundary, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 111–150. MR 1155228
Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI 10.2977/prims/1195171082

Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara