Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-17T22:13:37.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Vanishing theorems for the mod p cohomology of some simple Shimura varieties

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  03 November 2020

Teruhisa Koshikawa
Teruhisa Koshikawa*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto606-8502, Japan; E-mail: teruhisa@kurims.kyoto-u.ac.jp

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

Keywords

Hecke algebrasGalois representationsShimura varieties

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F80: Galois representations
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e38
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Boyer, Pascal, ‘Sur la torsion dans la cohomologie des variétés de Shimura de Kottwitz-Harris-Taylor’, J. Inst. Math. Jussieu 18 (2019), no. 3, 499–517, DOI 10.1017/s1474748017000093. MR3936639.CrossRefGoogle Scholar
Caraiani, Ana and Scholze, Peter, ‘On the generic part of the cohomology of compact unitary Shimura varieties’, Ann. of Math. (2) 186 (2017), no. 3, 649–766, DOI 10.4007/annals.2017.186.3.1. MR3702677CrossRefGoogle Scholar
Caraiani, Ana and Scholze, Peter, ‘On the generic part of the cohomology of non-compact unitary Shimura varieties’, https://arxiv.org/abs/1909.01898.Google Scholar
Emerton, Matthew and Gee, Toby, ‘p-adic Hodge-theoretic properties of étale cohomology with mod p coeffcients, and the cohomology of Shimura varieties’, Algebra Number Theory 9 (2015), no. 5, 1035–1088, DOI 10.2140/ant.2015.9.1035. MR3365999CrossRefGoogle Scholar
Fujiwara, Kazuhiro, ‘Galois deformations and arithmetic geometry of Shimura varieties’, in International Congress of Mathematicians Vol. II (Eur. Math. Soc., Zürich, 2006), 347–371.CrossRefGoogle Scholar
Guralnick, Robert and Malle, Gunter, ‘Characteristic polynomials and fixed spaces of semisimple elements, Recent developments in Lie algebras, groups and representation theory’, in Proc. Sympos. Pure Math., vol. 86 (Amer. Math. Soc., Providence, RI, 2012), 173–186, DOI 10.1090/pspum/086/1417. Google Scholar
Harris, Michael and Taylor, Richard, ‘The geometry and cohomology of some simple Shimura varieties’, in Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Lan, Kai-Wen and Suh, Junecue, ‘Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties’, Duke Math. J. 161 (2012), no. 6, 1113–1170, DOI 10.1215/00127094-1548452. CrossRefGoogle Scholar
Scott, Leonard L., ‘Matrices and cohomology’, Ann. of Math . (2) 105 (1977), no. 3, 473–492, DOI 10.2307/1970920. CrossRefGoogle Scholar
Shin, Sug Woo, ‘Supercuspidal part of the mod l cohomology of GU(1; n – 1)-Shimura varieties’, J. Reine Angew. Math. 705 (2015), 1–21, DOI 10.1515/crelle-2013-0057. CrossRefGoogle Scholar
Wedhorn, Torsten, ‘Congruence relations on some Shimura varieties’, J. Reine Angew. Math. 524 (2000), 43–71, DOI 10.1515/crll.2000.060. Google Scholar