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The stack of G-zips is a Mori dream space

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

Published online by Cambridge University Press:  19 September 2025

Jean-Stefan Koskivirta
Jean-Stefan Koskivirta*
Affiliation:
University of Caen Normandie , Department of Mathematics, Caen 14032, France
*
e-mail: jeanstefan.koskivirta@gmail.com
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Abstract

We first extend previous results of Koskivirta with Wedhorn and Goldring regarding the existence of $\mu $-ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of G-zips of Pink–Wedhorn–Ziegler. Then, we define and study the Cox ring of the stack of G-zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of vector-valued automorphic forms on the stack of G-zips and study its properties. We prove that it is finitely generated in certain cases.

Keywords

Stack of G-zipsMori dream spaceautomorphic forms

MSC classification

Primary: 14G35: Modular and Shimura varieties 20G40: Linear algebraic groups over finite fields

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported by JSPS KAKENHI Grant Number 21K13765 and by the University of Caen Normandie.

References

Andreatta, F., On two period maps: Ekedahl-Oort and fine Deligne-Lusztig stratifications . Math. Ann. 385(2023), nos. 1–2, 511–550.Google Scholar
Artin, M., Bertin, J. E., Demazure, M., Gabriel, P., Grothendieck, A., Raynaud, M., and Serre, J.-P., SGA3: Schémas en groupes. Vol. 1963/1964. Institut des Hautes Études Scientifiques, Paris, 1965/1966.Google Scholar
Borel, A., Linear algebraic groups. 2nd ed., Volume 126 of Graduate Texts in Mathematics, Springer-Verlag, New York, NY, 1991.Google Scholar
Bourbaki, N., Algèbre commutative, Springer, Berlin, Heidelberg, 2009. chapitres 5-7.Google Scholar
Ekedahl, T. and van der Geer, G., Cycle classes of the E-O stratification on the moduli of abelian varieties . In: Tschinkel, Y. and Zarhin, Y. (eds.), Algebra, arithmetic and geometry, volume 269 of Progress in Mathematics, Birkhäuser, Boston, MA, 2009, pp. 567–636.Google Scholar
Goldring, W. and Koskivirta, J.-S., Automorphic vector bundles with global sections on $G{\text{-}}Zip^{\mathcal{Z}}$ -schemes. Compos. Math. 154(2018), 2586–2605.Google Scholar
Goldring, W. and Koskivirta, J.-S., Strata Hasse invariants, Hecke algebras and Galois representations . Invent. Math. 217(2019), no. 3, 887–984.Google Scholar
Goldring, W. and Koskivirta, J.-S., Stratifications of flag spaces and functoriality . Int. Math. Res. Notices 2019(2019), no. 12, 3646–3682.Google Scholar
Goldring, W. and Koskivirta, J.-S., Divisibility of mod $p$ automorphic forms and the cone conjecture for certain Shimura varieties of Hodge-type. Preprint, 2022. arXiv:2211.16817.Google Scholar
Goldring, W. and Koskivirta, J.-S., Griffiths–Schmid conditions for automorphic forms via characteristic $p$ . Preprint, 2022. arXiv:2211.16819.Google 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
Hu, Y. and Keel, S., Mori dream spaces and GIT . Mich. Math. J. 48(2000), no. 1, 331–348.Google Scholar
Imai, N., Kato, H., and Youcis, A., The prismatic realization functor for Shimura varieties of abelian type, Preprint, 2023. arXiv:2310.08472.Google Scholar
Imai, N. and Koskivirta, J.-S., Automorphic vector bundles on the stack of $G$ -zips . Forum Math. Sigma. 9(2021), Article no. e37, 31 pp.Google Scholar
Imai, N. and Koskivirta, J.-S., Weights of mod $p$ automorphic forms and partial Hasse invariants. Preprint, 2022, with an appendix by W. Goldring. arXiv:2211.16207.Google Scholar
Imai, N. and Koskivirta, J.-S., Partial Hasse invariants for Shimura varieties of Hodge-type . Adv. Math. 440(2024), 109518.Google Scholar
Jantzen, J., Representations of algebraic groups. 2nd ed., volume 107 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2003.Google Scholar
Kisin, M., Integral models for Shimura varieties of abelian type . J. Am. Math. Soc. 23(2010), no. 4, 967–1012.Google Scholar
Knop, F., Kraft, H., Luna, D., and Vust, T., Local properties of algebraic group actions . In: Transformationsgruppen und Invariantentheorie, volume 13 of DMV Seminar, Birkhauser, Basel, 1989, pp. 63–75.Google Scholar
Knop, F., Kraft, H., and Vust, T., The Picard group of a  $G$ -variety. In: Transformationsgruppen und Invariantentheorie, volume 13 of DMV Seminar, Birkhauser, Basel, 1989, pp. 77–87.Google Scholar
Koskivirta, J.-S., Automorphic forms on the stack of $G$ -zips . Results Math. 74(2019), no. 3, Article no. 91, 52 pp.Google Scholar
Koskivirta, J.-S., Cohomology vanishing for Ekedahl-Oort strata on Hilbert-Blumenthal Shimura varieties. Preprint, 2023. arXiv:2311.18562.Google Scholar
Koskivirta, J.-S. and Wedhorn, T., Generalized $\mu$ -ordinary Hasse invariants . J. Algebra. 502(2018), 98–119.Google Scholar
Kottwitz, R., Points on some Shimura varieties over finite fields . J. Am. Math. Soc. 5(1992), no. 2, 373–444.Google Scholar
Moonen, B., Serre-Tate theory for moduli spaces of PEL-type . Ann. Sci. ENS. 37(2004), no. 2, 223–269.Google Scholar
Moonen, B. and Wedhorn, T., Discrete invariants of varieties in positive characteristic . Int. Math. Res. Notices 72(2004), 3855–3903.Google Scholar
Noether, E., Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik $p$ . Nachr. Ges. Wiss. Göttingen (1926), no. 1926, 28–35.Google 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 . Pac. J. Math. 274(2015), no. 1, 183–236.Google Scholar
Shen, X., Yu, C.-F., and Zhang, C., EKOR strata for Shimura varieties with parahoric level structure . Duke Math. J. 170(2021), no. 14, 3111–3236.Google Scholar
Springer, T., Linear algebraic groups. 2nd ed., volume 9 of Progress in Mathematics, Birkhauser, Boston, MA, 1998.Google Scholar
Vasiu, A., Integral canonical models of Shimura varieties of preabelian type . Asian J. Math. 3(1999), 401–518.Google Scholar
Wortmann, D., The $\mu$ -locus for Shimura varieties of Hodge type. Preprint, 2013. arXiv:1310.6444.Google Scholar
Zhang, C., Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type . Can. J. Math. 70(2018), no. 2, 451–480.Google Scholar