Note on weight-monodromy conjecture for 𝑝-adically uniformized varieties

Mieda, Yoichi

doi:10.1090/proc/14375

Note on weight-monodromy conjecture for $p$-adically uniformized varieties
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Abstract:

We prove the weight-monodromy conjecture for varieties which are $p$-adically uniformized by a product of the Drinfeld upper half-spaces. It is an easy consequence of Dat’s work on the cohomology complex of the Drinfeld upper half-space.

References

Jean François Dat, Espaces symétriques de Drinfeld et correspondance de Langlands locale, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 1, 1–74 (French, with English and French summaries). MR 2224658, DOI 10.1016/j.ansens.2005.11.002
J.-F. Dat, Théorie de Lubin-Tate non-abélienne et représentations elliptiques, Invent. Math. 169 (2007), no. 1, 75–152 (French, with English summary). MR 2308851, DOI 10.1007/s00222-007-0044-3
Pierre Deligne, Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 425–430 (French). MR 0441965
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307.
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252.
A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
V. G. Drinfel′d, Coverings of $p$-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29–40 (Russian). MR 0422290
Torsten Ekedahl, On the adic formalism, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 197–218. MR 1106899
Laurent Fargues, L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld et applications cohomologiques, L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld, Progr. Math., vol. 262, Birkhäuser, Basel, 2008, pp. 1–325 (French). MR 2441312
R. Huber, A comparison theorem for $l$-adic cohomology, Compositio Math. 112 (1998), no. 2, 217–235. MR 1626021, DOI 10.1023/A:1000345530725
Tetsushi Ito, Weight-monodromy conjecture for $p$-adically uniformized varieties, Invent. Math. 159 (2005), no. 3, 607–656. MR 2125735, DOI 10.1007/s00222-004-0395-y
Uwe Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), no. 2, 207–245. MR 929536, DOI 10.1007/BF01456052
Yoichi Mieda, Lefschetz trace formula for open adic spaces, J. Reine Angew. Math. 694 (2014), 85–128. MR 3259040, DOI 10.1515/crelle-2012-0103
Yoichi Mieda, Zelevinsky involution and $\ell$-adic cohomology of the Rapoport-Zink tower, preprint, arXiv:1401.5135, 2014.
M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), no. 1, 21–101 (German). MR 666636, DOI 10.1007/BF01394268
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
Peter Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313. MR 3090258, DOI 10.1007/s10240-012-0042-x
X. Shen, On the $\ell$-adic cohomology of some $p$-adically uniformized Shimura varieties, to appear in Journal of the Institute of Mathematics of Jussieu, https://doi.org/10.1017/S1474748016000360, 2016.
Richard Taylor and Teruyoshi Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), no. 2, 467–493. MR 2276777, DOI 10.1090/S0894-0347-06-00542-X
M.-F. Vignéras, Extensions between irreducible representations of a $p$-adic $\textrm {GL}(n)$, Pacific J. Math. (1997), no. Special Issue, 349–357, Olga Taussky-Todd: in memoriam.
Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656