The Langlands-Kottwitz method and deformation spaces of 𝑝-divisible groups

Scholze, Peter

doi:10.1090/S0894-0347-2012-00753-X

The Langlands-Kottwitz method and deformation spaces of $p$-divisible groups
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by Peter Scholze

J. Amer. Math. Soc. 26 (2013), 227-259

DOI: https://doi.org/10.1090/S0894-0347-2012-00753-X

Published electronically: August 16, 2012

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Abstract:

We extend the results of Kottwitz on points of Shimura varieties over finite fields to cases of bad reduction. The “test function” whose twisted orbital integrals appear in the final expression is defined geometrically using deformation spaces of $p$-divisible groups.

References

P. Berthelot. Cohomologie rigide et cohomologie rigide à support propre. http://perso.univ-rennes1.fr/pierre.berthelot/publis/Cohomologie_Rigide_I.pdf.
Gerd Faltings, Group schemes with strict $\scr O$-action, Mosc. Math. J. 2 (2002), no. 2, 249–279. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1944507, DOI 10.17323/1609-4514-2002-2-2-249-279
Laurent Fargues, Cohomologie des espaces de modules de groupes $p$-divisibles et correspondances de Langlands locales, Astérisque 291 (2004), 1–199 (French, with English and French summaries). Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales. MR 2074714
D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144 (2001), no. 2, 253–280. MR 1826370, DOI 10.1007/s002220100122
Ulrich Görtz, Topological flatness of local models in the ramified case, Math. Z. 250 (2005), no. 4, 775–790. MR 2180374, DOI 10.1007/s00209-005-0774-0
T. Haines and B. C. Ngô, Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), no. 2, 117–150. MR 1923579, DOI 10.1023/A:1019666710051
T. Haines and M. Rapoport. Shimura varieties with $\Gamma _1(p)$-level structure. in preparation.
Thomas J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 583–642. MR 2192017
Roland Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996. MR 1734903, DOI 10.1007/978-3-663-09991-8
R. Huber, A comparison theorem for $l$-adic cohomology, Compositio Math. 112 (1998), no. 2, 217–235. MR 1626021, DOI 10.1023/A:1000345530725
R. Huber, A finiteness result for direct image sheaves on the étale site of rigid analytic varieties, J. Algebraic Geom. 7 (1998), no. 2, 359–403. MR 1620118
R. Huber, A finiteness result for the compactly supported cohomology of rigid analytic varieties, J. Algebraic Geom. 7 (1998), no. 2, 313–357. MR 1620114
Luc Illusie, Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck), Astérisque 127 (1985), 151–198 (French). Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). MR 801922
R. E. Kottwitz. Isomorphism classes of elliptic curves within an isogeny class over a finite field, unpublished notes.
Robert E. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), no. 3, 287–300. MR 761308, DOI 10.1007/BF01450697
Robert E. Kottwitz, Shimura varieties and $\lambda$-adic representations, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 161–209. MR 1044820
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
Kai-Wen Lan, Arithmetic compactifications of PEL-type Shimura varieties, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Harvard University. MR 2711676
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, 1973, pp. 361–500. MR 0354617
Yoichi Mieda, On $l$-independence for the étale cohomology of rigid spaces over local fields, Compos. Math. 143 (2007), no. 2, 393–422. MR 2309992, DOI 10.1112/S0010437X06002582
G. Pappas and M. Rapoport, Local models in the ramified case. I. The EL-case, J. Algebraic Geom. 12 (2003), no. 1, 107–145. MR 1948687, DOI 10.1090/S1056-3911-02-00334-X
G. Pappas and W. Zhu. Local models of Shimura varieties and a conjecture of Kottwitz. arXiv:1110.5588.
M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439, DOI 10.1515/9781400882601
P. Scholze. The Langlands-Kottwitz approach for some simple Shimura varieties. arXiv:1003.2451, to appear in Invent. Math.
P. Scholze. A new approach to the local Langlands correspondence for $\mathrm {GL}_n$ over $p$-adic fields. arXiv:1010.1540, to appear in Invent. Math.
Peter Scholze, The Langlands-Kottwitz approach for the modular curve, Int. Math. Res. Not. IMRN 15 (2011), 3368–3425. MR 2822177, DOI 10.1093/imrn/rnq225
P. Scholze and S. W. Shin. On the cohomology of compact unitary group Shimura varieties at ramified split places. arXiv:1110.0232.
Sug Woo Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), no. 3, 1645–1741. MR 2800722, DOI 10.4007/annals.2011.173.3.9
Yakov Varshavsky, Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara, Geom. Funct. Anal. 17 (2007), no. 1, 271–319. MR 2306659, DOI 10.1007/s00039-007-0596-9
Torsten Wedhorn, The dimension of Oort strata of Shimura varieties of PEL-type, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 441–471. MR 1827029