zbMATH — the first resource for mathematics

Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties. (Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples.) (French) Zbl 1172.14016
Let \(K\) be a finite extension of \({\mathbb Q}_p\) with ring of integers \({\mathcal O}_K\). For some fixed positive integer \(d\), we consider the group \(D^\times_{K,d}\) of invertibles of the central division algebra over \(K\) with invariant \(\frac{1}{d}\) an the Weil group \(W_K\) of \(K\).
Using étale cohomology, Deligne has constructed a series of representations \({\mathcal U}^i_{K,l,d}\) of the group \(GDW_K(d):=GL_d(K)\times D^\times_{K,d}\times W_K\). Now let \(\rho\) be any irreducible representation of \(D^\times_{K,d}\) such that the corresponding \(\pi:=JL(\rho)\) is a cuspidal representation of \(GL_d(K)\). The first result of the paper under review calculates the \(\rho\)-isotypical components \({\mathcal U}^i_{K,l,d}(\rho)\) of \({\mathcal U}^i_{K,l,d}\) completely via globalizing the investigations of [M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Annals of Mathematics Studies 151. (Princeton), NJ: Princeton University Press. (2001; Zbl 1036.11027)]: There, the authors study Shimura varieties of PEL type defined over some CM field F, which are associated to a particular group \(G/{\mathbb Q}\) of similitudes. To certain subgroups \(I\subset G({\mathbb A}^\infty)\) one associates a tower of Shimura varieties \(X_{\mathcal I}:=(X_I)_{I\in{\mathcal I}}\to Spec({\mathcal O}_v)\), where \(v\) is some prime above \(p\) such that the completion \(F_v\) of the localization of \(F\) in \(v\) is isomorphic to \(K\), on which \(G({\mathbb A}^\infty)\) acts via correspondences.
The special geometric fiber \(\overline{X}_{\mathcal I}\) of \(X_{\mathcal I}\) is stratified by locally closed Hecke schemes \(\overline{X}_{\mathcal I}^{=h}\) for \(1\leq d\leq h\). In the present paper, methods of Harris and Taylor to describe the restriction of the sheaf of vanishing cycles to these strata are generalized.
Now let \(\Psi_{\mathcal I}:=R\Psi_{\eta_v}({\overline{\mathbb Q}}_l)[d-1](\frac{d-1}{2})\) be the complex of vanishing cycles on \(\overline{X}_{\mathcal I}\) viewed as a complex of \(W_v\)-perverse Hecke sheaves, and for an irreducible cuspidal representation \(\pi_v\) of \(GL_g(F_v)\) denote \(\Psi_{{\mathcal I},\pi_v}\) the corresponding isotypical component. The second main result of the present article describes its bi-graded pieces with respect to kernels and images of the pro-nilpotent monodromy operator \(N\) in the category of \(W_{F_v}\)-perverse sheaves on \(\overline{X}_{\mathcal I}\).
The proof works by induction on the \({\mathcal U}^i_{F_v,l,h}\) for \(1\leq h < d\) (supposed to be known) of the Deligne-Carayol model. The initial step is comprised by former work of the author [P. Boyer, Invent. Math. 138, No. 3, 573–629 (1999; Zbl 1161.11408)]. The comparison theorem of Berkovich and Fargues is then applied in order to obtain the result except for those perverse sheaves with support in supersingular points. This case finally is treated using methods from Harris and Talyor’s work again in connection with the auto-duality of the local model with respect to the Zelevinski involution.
The article is nicely written, and also includes an appendix on Hecke schemes and their associated sheaves for the readers’ convenience.

14G22 Rigid analytic geometry
14G35 Modular and Shimura varieties
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G35 Varieties over global fields
11R39 Langlands-Weil conjectures, nonabelian class field theory
14L05 Formal groups, \(p\)-divisible groups
11G45 Geometric class field theory
11Fxx Discontinuous groups and automorphic forms
Full Text: DOI
[1] Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I, Luminy, 1981. Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982) · Zbl 0536.14011
[2] Boyer, P.: Cohomologie des systèmes locaux de Harris-Taylor et applications. http://people.math.jussieu.fr/\(\sim\)boyer/fichiers/MP-cohomologique.pdf
[3] Boyer, P.: Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale. Invent. Math. 138(3), 573–629 (1999) · Zbl 1161.11408 · doi:10.1007/s002220050354
[4] Boyer, P.: Faisceaux pervers des cycles évanescents des variétés de Drinfeld et groupes de cohomologies du modèle de Deligne-Carayol. http://people.math.jussieu.fr/\(\sim\)boyer/fichiers/livre.pdf , à paraître aux Mémoires de la SMF, 2009 · Zbl 1186.14003
[5] Carayol, H.: Sur les représentations l-adiques associées aux formes modulaires de Hilbert. Ann. Sci. Ec. Norm. Super. (4) 19, 409–468 (1986) · Zbl 0616.10025
[6] Carayol, H.: Nonabelian Lubin-Tate theory. In: Automorphic forms, Shimura Varieties, and L-Functions. Perspect. Math., vol. 11, pp. 15–39. Academic Press, Boston (1990)
[7] Dat, J.-F.: Théorie de Lubin-Tate non-abélienne et représentations elliptiques. Invent. Math. 169(1), 75–152 (2007) · Zbl 1180.22020 · doi:10.1007/s00222-007-0044-3
[8] Faltings, G.: A relation between two moduli spaces studied by V.G. Drinfeld. In: Algebraic Number Theory and Algebraic Geometry. Contemp. Math., vol. 300, pp. 115–129. Am. Math. Soc., Providence (2002) · Zbl 1062.14059
[9] Fargues, L.: Filtration de monodromie et cycles évanescents formels. Invent. Math. (2009). doi: 10.1007/s00222-009-0184-8 · Zbl 1182.14015
[10] Fargues, L.: Dualité de Poincaré et involution de Zelevinsky dans la cohomologie étale équivariante des espaces analytiques rigides. http://www.math.u-psud.fr/\(\sim\)fargues/Dualite.dvi , 2006
[11] Fargues, L., Genestier, A., Lafforgue, V.: L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld. Progress in Mathematics, vol. 262. Birkhäuser, Basel (2008), xii, 406 p. · Zbl 1136.14001
[12] Goertz, U., Haines, T.: The Jordan-Hoelder series for nearby cycles on some Shimura varieties and affine flag varieties. J. Reine Angew. Math. 609, 161–213 (2007) · Zbl 1157.14013 · doi:10.1515/CRELLE.2007.063
[13] Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001) · Zbl 1036.11027
[14] Henniart, G.: Sur la conjecture de Langlands locale pour GL n . J. Théor. Nr. Bord. 13(1), 167–187 (2001). 21st Journées Arithmétiques (Rome, 2001) · Zbl 1048.11093
[15] Ito, T.: Hasse invariants for somme unitary Shimura varieties. Math. Forsch. Oberwolfach report 28/2005, pp. 1565–1568 (2005)
[16] Kottwitz, R.E.: Stable trace formula: elliptic singular terms. Math. Ann. 275(3), 365–399 (1986) · Zbl 0591.10020 · doi:10.1007/BF01458611
[17] Kottwitz, R.E.: Points on some Shimura varieties over finite fields. J. Am. Math. Soc. 5(2), 373–444 (1992) · Zbl 0796.14014 · doi:10.1090/S0894-0347-1992-1124982-1
[18] Serre, J.-P.: Corps locaux, 2ème édition. Publications de l’Université de Nancago, No. VIII. Hermann, Paris (1968).
[19] Taylor, R., Yoshida, T.: Compatibility of local and global Langlands correspondences. J. Am. Math. Soc. 20, 467–493 (2007) · Zbl 1210.11118 · doi:10.1090/S0894-0347-06-00542-X
[20] Zelevinsky, A.V.: Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. Ec. Norm. Sup. (4) 13(2), 165–210 (1980) · Zbl 0441.22014
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.