Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T11:03:03.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Automorphic vector bundles on the stack of G-zips

Part of: Arithmetic problems. Diophantine geometry Linear algebraic groups and related topics

Published online by Cambridge University Press:  03 May 2021

Naoki Imai  and
Jean-Stefan Koskivirta
Naoki Imai
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan; E-mail: [email protected].
Jean-Stefan Koskivirta
Affiliation:
Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama 338-8570, Japan; E-mail: [email protected].

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.

For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

MSC classification

Primary: 14G35: Modular and Shimura varieties
Secondary: 20G40: Linear algebraic groups over finite fields
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e37
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), 2021. Published by Cambridge University Press

References

Artin, M., Bertin, J. E., Demazure, M., Gabriel, P., Grothendieck, A., Raynaud, M. and Serre, J.-P., SGA3: Schémas en groupes [Group schemes], Vol. 1963/64 (Institut des Hautes Études Scientifiques, Paris, 1965/1966).Google Scholar
Deligne, P., ‘Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques’ [Shimura varieties: Moduli interpretation and techniques of construction of canonical models], in Automorphic Forms, Representations and $L$-functions, Part 2, ed. by A. Borel and W. Casselman, Vol. 33 of Proc. Symp. Pure Math. (American Mathematical Society, Providence, RI, 1979), 247–289.CrossRefGoogle Scholar
Donkin, S., Rational Representations of Algebraic Groups: Tensor Products and Filtration, Vol. 1140 of Lecture Notes in Mathematics (Springer, Berlin, 1985).CrossRefGoogle Scholar
Goldring, W., Imai, N. and Koskivirta, J.-S., ‘Weights of mod $p$ automorphic forms and partial Hasse invariants’, Preprint, 2021.Google Scholar
Goldring, W. and Koskivirta, J.-S., ‘Automorphic vector bundles with global sections on $G$-Zip ${}^{{\mathcal{Z}}}$-schemes’, Compositio Math. 154 (2018), 2586–2605.CrossRefGoogle Scholar
Goldring, W. and Koskivirta, J.-S., ‘Strata Hasse invariants, Hecke algebras and Galois representations’, Invent. Math. 217(3) (2019), 887–984.CrossRefGoogle Scholar
Goldring, W. and Koskivirta, J.-S., ‘Zip stratifications of flag spaces and functoriality’, IMRN (2019) (12) (2019), 3646–3682.CrossRefGoogle Scholar
Goldring, W. and Nicole, M.-H., ‘The $\mu$-ordinary Hasse invariant of unitary Shimura varieties’, J. Reine Angew. Math. 728 (2017), 137–151.Google Scholar
Imai, N. and Koskivirta, J.-S., ‘Partial Hasse invariants for Shimura varieties of Hodge-type’, Preprint, 2021.Google Scholar
Jantzen, J., Representations of Algebraic Groups, Vol. 107 of Math. Surveys and Monographs, 2nd ed. (American Mathematical Society, Providence, RI, 2003).Google Scholar
Kim, W., ‘Rapoport–Zink uniformization of Shimura varieties’, Forum Math. Sigma 6 (2018), e16.CrossRefGoogle Scholar
Kisin, M., ‘Integral models for Shimura varieties of abelian type’, J. Amer. Math. Soc. 23(4) (2010), 967–1012.CrossRefGoogle Scholar
Koskivirta, J.-S., Normalization of closed Ekedahl-Oort strata, Can. Math. Bull. 61 3 (2018), 572–587.CrossRefGoogle Scholar
Koskivirta, J.-S., ‘Automorphic forms on the stack of $G$-zips’, Results Math. 74 (2019), no. 3, Paper No. 91, 52 pp.CrossRefGoogle Scholar
Koskivirta, J.-S. and Wedhorn, T., ‘Generalized $\mu$-ordinary Hasse invariants’, J. Algebra 502 (2018), 98–119.CrossRefGoogle Scholar
Kottwitz, R. E., ‘Shimura varieties and twisted orbital integrals’, Math. Ann. 269 (1984), 287–300.CrossRefGoogle Scholar
Milne, J., ‘Canonical models of (mixed) Shimura varieties and automorphic vector bundles’, in Automorphic Forms, Shimura Varieties, and $L$-Functions, Vol. I, ed. by Clozel, L. and Milne, J., Vol. 11 of Perspect. Math. (Academic Press, Boston, 1990), 283–414.Google Scholar
Moonen, B., ‘Serre-Tate theory for moduli spaces of PEL-type’, Ann. Sci. ENS 37(2) (2004), 223–269.Google Scholar
Moonen, B. and Wedhorn, T., ‘Discrete invariants of varieties in positive characteristic’, IMRN 72 (2004), 3855–3903.CrossRefGoogle Scholar
Pink, R., Wedhorn, T. and Ziegler, P., ‘Algebraic zip data’, Doc. Math. 16 (2011), 253–300.Google Scholar
Pink, R., Wedhorn, T. and Ziegler, P., ‘$F$-zips with additional structure’, Pacific J. Math. 274(1) (2015), 183–236.CrossRefGoogle Scholar
Shen, X., Yu, C.-F. and Zhang, C., ‘EKOR strata for Shimura varieties with parahoric level structure’, Preprint, 2019, arXiv:1910.07785.Google Scholar
Springer, T., Linear Algebraic Groups, Vol. 9 of Progress in Math., 2nd edn. (Birkhauser, Boston, MA, 1998).CrossRefGoogle Scholar
Urban, E., ‘Nearly overconvergent modular forms’, in Iwasawa Theory, Vol. 7 of Contrib. Math. Comput. Sci. (Springer, Heidelberg, Germany, 2014), 401–441.CrossRefGoogle Scholar
Vasiu, A., ‘Integral canonical models of Shimura varieties of preabelian type’, Asian J. Math. 3 (1999), 401–518.CrossRefGoogle Scholar
Wortmann, D., ‘The $\mu$-ordinary locus for Shimura varieties of Hodge type’, Preprint, 2013, arXiv:1310.6444.Google Scholar
Xiao, L. and Zhu, X., ‘Cycles on Shimura varieties via geometric Satake’, Preprint, 2017, arXiv:1707.05700.Google Scholar
Xiao, L. and Zhu, X., ‘On vector-valued twisted conjugation invariant functions on a group’, in Representations of Reductive Groups, Vol. 101 of Proc. Sympos. Pure Math. (American Mathematical Society, Providence, RI, 2019), 361–425. With an appendix by Stephen Donkin.CrossRefGoogle Scholar
Zhang, C., ‘Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type’, Can. J. Math. 70(2) (2018), 451–480.CrossRefGoogle Scholar