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

Morel, Sophie^1, 2

¹ Laboratoire de mathématique, Université Paris-Sud, bâtiment 425, 91405 Orsay Cedex, France
² After September 1, 2006: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540

Journal of the American Mathematical Society, Tome 21 (2008) no. 1, pp. 23-61

Résumé

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.

Détail

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

Bibliographie

[1] Ash, A., Mumford, D., Rapoport, M., Tai, Y. Smooth compactification of locally symmetric varieties 1975

[2] Baily, W. L., Jr., Borel, A. Compactification of arithmetic quotients of bounded symmetric domains Ann. of Math. (2) 1966 442 528

[3] Beä­Linson, A. A. On the derived category of perverse sheaves 1987 27 41

[4] Analyse et topologie sur les espaces singuliers. I 1982 172

[5] Brylinski, J.-L., Labesse, J.-P. Cohomologie d’intersection et fonctions 𝐿 de certaines variétés de Shimura Ann. Sci. École Norm. Sup. (4) 1984 361 412

[6] Faltings, Gerd, Chai, Ching-Li Degeneration of abelian varieties 1990

[7] Deligne, Pierre La conjecture de Weil. II Inst. Hautes Études Sci. Publ. Math. 1980 137 252

[8] Fujiwara, Kazuhiro Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture Invent. Math. 1997 489 533

[9] Goresky, M., Harder, G., Macpherson, R. Weighted cohomology Invent. Math. 1994 139 213

[10] Kottwitz, Robert E. Points on some Shimura varieties over finite fields J. Amer. Math. Soc. 1992 373 444

[11] Kottwitz, Robert E., Rapoport, Michael Contribution of the points at the boundary 1992 111 150

[12] Langlands, R. P. Modular forms and ℓ-adic representations 1973 361 500

[13] Levine, Marc Tate motives and the vanishing conjectures for algebraic 𝐾-theory 1993 167 188

[14] The zeta functions of Picard modular surfaces 1992

[15] Looijenga, E., Rapoport, M. Weights in the local cohomology of a Baily-Borel compactification 1991 223 260

[16] Pink, Richard Arithmetical compactification of mixed Shimura varieties 1990

[17] Pink, Richard On 𝑙-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification Math. Ann. 1992 197 240

[18] Pink, Richard On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne Ann. of Math. (2) 1992 483 525

[19] Kottwitz, Robert E., Rapoport, Michael Contribution of the points at the boundary 1992 111 150

[20] Saito, Morihiko Mixed Hodge modules Publ. Res. Inst. Math. Sci. 1990 221 333