Sur une conjecture de Kottwitz au bord

Stroh, Benoît

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 1, pp. 143-165.

Nous nous intéressons à la cohomologie d'intersection de la compactification minimale des variétés de Siegel à certaines places de mauvaise réduction. Nous calculons la trace semi-simple du morphisme de Frobenius sur les fibres des cycles proches du complexe d'intersection. Nous obtenons une généralisation commune de résultats de Morel et de Haines et Ngô.

We are interested in the intersection cohomology of the minimal compactification of Siegel modular varieties at some places of bad reduction. We compute the semi-simple trace of the Frobenius morphism on the fibers of the nearby cycles of the intersection complex. We obtain a common generalization of results of Morel, Haines and Ngô.

[1] A. A. Beĭlinson, On the derived category of perverse sheaves1984-1986), Lecture Notes in Math. 1289, Springer, 1987, 27-41. | Zbl

[2] A. A. Beĭlinson, J. Bernstein & P. Deligne, Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, 1982, 5-171. | MR | Zbl

[3] G. Faltings & C.-L. Chai, Degeneration of abelian varieties, Ergebnisse Math. Grenzg. (3) 22, Springer, 1990. | Zbl

[4] A. Genestier & J. Tilouine, Systèmes de Taylor-Wiles pour ${\mathrm{GSp}}_4$, Astérisque 302 (2005), 177-290. | Numdam | Zbl

[5] U. Görtz & T. J. Haines, The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties, J. reine angew. Math. 609 (2007), 161-213. | MR | Zbl

[6] T. J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4, Amer. Math. Soc., 2005, 583-642. | MR | Zbl

[7] T. J. Haines & B. C. Ngô, Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), 117-150. | MR | Zbl

[8] L. Illusie, Autour du théorème de monodromie locale, Astérisque 223 (1994), 9-57. | Numdam | MR | Zbl

[9] R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373-444. | MR | Zbl

[10] R. E. Kottwitz & M. Rapoport, Minuscule alcoves for ${\mathrm{GL}}_n$ and $G{\mathrm{Sp}}_{2n}$, Manuscripta Math. 102 (2000), 403-428. | MR | Zbl

[11] K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, Thèse, Harvard University, 2008. | MR

[12] S. Morel, Complexes d'intersection des compactifications de Baily-Borel 2005.

[13] S. Morel, Complexes pondérés sur les compactifications de Baily-Borel : le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23-61. | MR | Zbl

[14] S. Morel, On the cohomology of certain noncompact Shimura varieties, Annals of Math. Studies 173, Princeton Univ. Press, 2010. | MR | Zbl

[15] S. Morel, Cohomologie d'intersection des variétés modulaires de Siegel, suite, à paraître dans Compositio Math. | Zbl

[16] B. C. Ngô & A. Genestier, Alcôves et $p$-rang des variétés abéliennes, Ann. Inst. Fourier (Grenoble) 52 (2002), 1665-1680. | Numdam | MR | Zbl

[17] R. Pink, On $l$-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification, Math. Ann. 292 (1992), 197-240. | MR | Zbl

[18] M. Rapoport, On the bad reduction of Shimura varieties 1988), Perspect. Math. 11, Academic Press, 1990, 253-321. | MR | Zbl

[19] J-P. Serre, Zeta and $L$ functions, in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, 1965, 82-92. | MR | Zbl

[20] B. Stroh, Compactification de variétés de Siegel aux places de mauvaise réduction, Bull. Soc. Math. France 138 (2010), 259-315. | Numdam | MR | Zbl

[21] B. Stroh, Compactification minimale et mauvaise réduction, Ann. Inst. Fourier (Grenoble) 60 (2010), 1035-1055. | Numdam | MR | Zbl