Partial Hasse invariants for Shimura varieties of Hodge-type. (English) Zbl 1539.14047
Partial Hasse invariants were originally defined in [E. Z. Goren, Isr. J. Math. 122, 157–174 (2001; Zbl 1066.11018)] and [F. Andreatta and E. Z. Goren, Hilbert modular forms: mod \(p\) and \(p\)-adic aspects. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1071.11023)] as certain Hilbert modular forms in positive characteristic. They are associated to codimension one Ekedahl-Oort strata of Hilbert modular varieties over finite fields. These invariants have later been generalised by several authors using crystalline cohomology. The present paper goes in a different direction and provides a general theory of partial Hasse invariants in the framework of \(G\)-zips.
Let \(G\) be a connected reductive group over a finite field \(\mathbb{F}_q\) and let \(\mu\) be a geometric cocharacter of \(G\). To the pair \((G,\mu)\), or rather to the “zip datum” uniquely determined by \((G,\mu)\), Pink, Wedhorn and Ziegler associated a stack \(G\text{-Zip}^\mu\) classifying \(G\)-zips “of type \(\mu\)”. It is a quotient stack of \(G\) over an algebraic closure of \(\mathbb{F}_q\), which admits a stratification \(G\text{-Zip}^\mu=\bigsqcup_w\mathcal{X}_w\) indexed by a subset of the Weyl group of \(G\). If the pair \((G,\mu)\) arises from a Shimura datum of Hodge type with good reduction at \(p\), then there exists a smooth map \(\zeta\colon S_K\to G\text{-Zip}^\mu\), where \(S_K\) is the special fibre of an integral model of the associated Shimura variety (see [C. Zhang, Can. J. Math. 70, No. 2, 451–480 (2018; Zbl 1423.14180)]). The preimage via this map of the above-mentioned stratification of \(G\text{-Zip}^\mu\) coincides with the Ekedahl-Oort stratification of \(S_K\).
A zip partial Hasse invariant is defined by the authors as a section of some “automorphic” line bundle over \(G\text{-Zip}^\mu\), whose vanishing locus is the Zariski closure of a codimension one stratum \(\mathcal{X}_w\) of \(G\text{-Zip}^\mu\). In the Shimura case, the previous notions of partial Hasse invariants are recovered via the map \(\zeta\); several examples are explicitely computed in this setup. The authors introduce at the same time the related notion of flag partial Hasse invariants on a stack \(G\text{-ZipFlag}^\mu\) projecting onto \(G\text{-Zip}^\mu\). This refined framework allows them to construct certain Verschiebung homomorphisms of automorphic vector bundles. If the starting datum is defined over \(\mathbb{F}_q\), then these homomorphisms yield a factorisation of partial Hasse invariants. Another important result of the paper is the construction of flag partial Hasse invariants which are primitive, in the sense that they belong to the “socle” of the automorphic vector bundle at hand.
The authors apply the outcomes of this paper in a follow-up work [W. Goldring et al., “Weights of mod \(p\) automorphic forms and partial Hasse invariants”, Preprint, arXiv:2211.16207], in order to generalise results from [F. Diamond and P. L Kassaei, Compos. Math. 153, No. 9, 1769–1778 (2017; Zbl 1391.11073)] and [F. Diamond and P. L. Kassaei, Int. Math. Res. Not. 2023, No. 14, 12148–12171 (2023; Zbl 1537.11062)].
Let \(G\) be a connected reductive group over a finite field \(\mathbb{F}_q\) and let \(\mu\) be a geometric cocharacter of \(G\). To the pair \((G,\mu)\), or rather to the “zip datum” uniquely determined by \((G,\mu)\), Pink, Wedhorn and Ziegler associated a stack \(G\text{-Zip}^\mu\) classifying \(G\)-zips “of type \(\mu\)”. It is a quotient stack of \(G\) over an algebraic closure of \(\mathbb{F}_q\), which admits a stratification \(G\text{-Zip}^\mu=\bigsqcup_w\mathcal{X}_w\) indexed by a subset of the Weyl group of \(G\). If the pair \((G,\mu)\) arises from a Shimura datum of Hodge type with good reduction at \(p\), then there exists a smooth map \(\zeta\colon S_K\to G\text{-Zip}^\mu\), where \(S_K\) is the special fibre of an integral model of the associated Shimura variety (see [C. Zhang, Can. J. Math. 70, No. 2, 451–480 (2018; Zbl 1423.14180)]). The preimage via this map of the above-mentioned stratification of \(G\text{-Zip}^\mu\) coincides with the Ekedahl-Oort stratification of \(S_K\).
A zip partial Hasse invariant is defined by the authors as a section of some “automorphic” line bundle over \(G\text{-Zip}^\mu\), whose vanishing locus is the Zariski closure of a codimension one stratum \(\mathcal{X}_w\) of \(G\text{-Zip}^\mu\). In the Shimura case, the previous notions of partial Hasse invariants are recovered via the map \(\zeta\); several examples are explicitely computed in this setup. The authors introduce at the same time the related notion of flag partial Hasse invariants on a stack \(G\text{-ZipFlag}^\mu\) projecting onto \(G\text{-Zip}^\mu\). This refined framework allows them to construct certain Verschiebung homomorphisms of automorphic vector bundles. If the starting datum is defined over \(\mathbb{F}_q\), then these homomorphisms yield a factorisation of partial Hasse invariants. Another important result of the paper is the construction of flag partial Hasse invariants which are primitive, in the sense that they belong to the “socle” of the automorphic vector bundle at hand.
The authors apply the outcomes of this paper in a follow-up work [W. Goldring et al., “Weights of mod \(p\) automorphic forms and partial Hasse invariants”, Preprint, arXiv:2211.16207], in order to generalise results from [F. Diamond and P. L Kassaei, Compos. Math. 153, No. 9, 1769–1778 (2017; Zbl 1391.11073)] and [F. Diamond and P. L. Kassaei, Int. Math. Res. Not. 2023, No. 14, 12148–12171 (2023; Zbl 1537.11062)].
Reviewer: Andrea Marrama (Padova)
References:
| [1] | Artin, M.; Bertin, J. E.; Demazure, M.; Gabriel, P.; Grothendieck, A.; Raynaud, M.; Serre, J.-P., SGA3: Schémas en groupes, vol. 1963/64, , (, 1, 9, 6, 5, /, 1, 9, 6, 6, ), Institut des Hautes Études Scientifiques: Institut des Hautes Études Scientifiques Paris · Zbl 0212.52809 |
| [2] | Andreatta, F.; Goren, E., Hilbert Modular Forms: mod p and p-adic Aspects, Mem. Amer. Math Soc., vol. 173, (2005), Amer. Math. Soc. · Zbl 1071.11023 |
| [3] | Bijakowski, S.; Hernandez, V., Groupes p-divisibles avec condition de Pappas-Rapoport et invariants de Hasse, J. Éc. Polytech. Math., 4, 935-972, (2017) · Zbl 1423.14256 |
| [4] | Bijakowski, S., Partial Hasse invariants, partial degrees, and the canonical subgroup, Can. J. Math., 70, 4, 742-772, (2018) · Zbl 1446.11105 |
| [5] | Deligne, P., Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques, (Borel, A.; Casselman, W., Automorphic Forms, Representations and L-Functions, Part 2, Proc. Symp. Pure Math., vol. 33, (1979), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI). (Borel, A.; Casselman, W., Automorphic Forms, Representations and L-Functions, Part 2, Proc. Symp. Pure Math., vol. 33, (1979), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), Proc. Sympos. Pure Math., 247-289, (1977), Oregon State Univ.: Oregon State Univ. Corvallis, OR · Zbl 0437.14012 |
| [6] | Diamond, F.; Kassaei, P., Minimal weights of Hilbert modular forms in characteristic p, Compos. Math., 153, 9, 1769-1778, (2017) · Zbl 1391.11073 |
| [7] | Diamond, F.; Kassaei, P. L., The cone of minimal weights for mod p Hilbert modular forms, Int. Math. Res. Not., 14, 12148-12171, (2023) · Zbl 1537.11062 |
| [8] | Donkin, S., Rational Representations of Algebraic Groups: Tensor Products and Filtration, Lect. Notes in Math., vol. 1140, (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0586.20017 |
| [9] | Garibaldi, S.; Guralnick, R. M.; Nakano, D. K., Globally irreducible Weyl modules, J. Algebra, 477, 69-87, (2017) · Zbl 1427.20062 |
| [10] | Goldring, W.; Koskivirta, J.-S., Automorphic vector bundles with global sections on G-Zip \({}^{\mathcal{Z}} \)-schemes, Compos. Math., 154, 2586-2605, (2018), MR 4582533 · Zbl 1457.11045 |
| [11] | Goldring, W.; Koskivirta, J.-S., Strata Hasse invariants, Hecke algebras and Galois representations, Invent. Math., 217, 3, 887-984, (2019) · Zbl 1431.11071 |
| [12] | Goldring, W.; Koskivirta, J.-S., Stratifications of flag spaces and functoriality, Int. Math. Res. Not., 2019, 12, 3646-3682, (2019) · Zbl 1432.14022 |
| [13] | Goldring, W.; Nicole, M.-H., The μ-ordinary Hasse invariant of unitary Shimura varieties, J. Reine Angew. Math., 728, 137-151, (2017) · Zbl 1391.14045 |
| [14] | Goren, E., Hasse invariants for Hilbert modular varieties, Isr. J. Math., 122, 157-174, (2001) · Zbl 1066.11018 |
| [15] | Griffiths, P.; Schmid, W., Locally homogeneous complex manifolds, Acta Math., 123, 253-302, (1969) · Zbl 0209.25701 |
| [16] | Hernandez, V., Invariants de Hasse μ-ordinaires, Ann. Inst. Fourier, 68, 4, 1519-1607, (2018) · Zbl 1423.14145 |
| [17] | He, X.; Nie, S., On the μ-ordinary locus of a Shimura variety, Adv. Math., 321, 513-528, (2017) · Zbl 1410.11069 |
| [18] | Imai, N.; Koskivirta, J.-S., Automorphic vector bundles on the stack of G-zips, Forum Math. Sigma, 9, Article e37 pp., (2021), 31 pp. · Zbl 1492.14041 |
| [19] | Imai, N.; Koskivirta, J.-S., Weights of mod p automorphic forms and partial Hasse invariants, (2022), with an appendix by W. Goldring, Preprint |
| [20] | Jantzen, J., Representations of Algebraic Groups, Math. Surveys and Monographs, vol. 107, (2003), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1034.20041 |
| [21] | Kisin, M., Integral models for Shimura varieties of abelian type, J. Am. Math. Soc., 23, 4, 967-1012, (2010) · Zbl 1280.11033 |
| [22] | Knop, F.; Kraft, H.; Luna, D.; Vust, T., Local properties of algebraic group actions, (Transformationsgruppen und Invariantentheorie. Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, (1989), Birkhäuser), 63-75 · Zbl 0722.14032 |
| [23] | Koskivirta, J.-S., Automorphic forms on the stack of G-zips, Results Math., 74, 3, Article 91 pp., (2019), 52 pp. · Zbl 07076898 |
| [24] | Koskivirta, J.-S.; Wedhorn, T., Generalized μ-ordinary Hasse invariants, J. Algebra, 502, 98-119, (2018) · Zbl 1401.11109 |
| [25] | Lan, K.-W.; Stroh, B., Compactifications of subschemes of integral models of Shimura varieties, Forum Math. Sigma, 6, Article e18 pp., (2018), 105 pp. · Zbl 1454.11115 |
| [26] | Milne, J., Canonical models of (mixed) Shimura varieties and automorphic vector bundles, (Clozel, L.; Milne, J., Automorphic Forms, Shimura Varieties, and L-Functions, Vol. I, Perspect. Math., vol. 11, (1990), Academic Press: Academic Press Boston, MA), 283-414, Proc. Conf. held in Ann Arbor, MI, 1988 · Zbl 0704.14016 |
| [27] | Moonen, B., Serre-Tate theory for moduli spaces of PEL-type, Ann. Sci. ENS, 37, 2, 223-269, (2004) · Zbl 1107.11028 |
| [28] | Pink, R.; Wedhorn, T.; Ziegler, P., Algebraic zip data, Doc. Math., 16, 253-300, (2011) · Zbl 1230.14070 |
| [29] | Pink, R.; Wedhorn, T.; Ziegler, P., F-zips with additional structure, Pac. J. Math., 274, 1, 183-236, (2015) · Zbl 1349.14077 |
| [30] | Rosenlicht, M., Toroidal algebraic groups, Proc. Am. Math. Soc., 12, 984-988, (1961) · Zbl 0107.14703 |
| [31] | Ramanan, S.; Ramanathan, A., Projective normality of flag varieties and Schubert varieties, Invent. Math., 79, 217-224, (1985) · Zbl 0553.14023 |
| [32] | Springer, T., Linear Algebraic Groups, Progress in Math., vol. 9, (1998), Birkhäuser · Zbl 0927.20024 |
| [33] | Steinberg, R., Representations of algebraic groups, Nagoya Math. J., 22, 33-56, (1963) · Zbl 0271.20019 |
| [34] | Shen, X.; Yu, C.-F.; Zhang, C., EKOR strata for Shimura varieties with parahoric level structure, Duke Math. J., 170, 14, 3111-3236, (2021) · Zbl 1481.14049 |
| [35] | Vasiu, A., Integral canonical models of Shimura varieties of preabelian type, Asian J. Math., 3, 401-518, (1999) · Zbl 1002.11052 |
| [36] | Wedhorn, T., Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. ENS, 32, 5, 575-618, (1999) · Zbl 0983.14024 |
| [37] | Wedhorn, T., Bruhat strata and F-zips with additional structure, Münster J. Math., 7, 2, 529-556, (2014) · Zbl 1375.14083 |
| [38] | Wortmann, D., The μ-ordinary locus for Shimura varieties of Hodge type, (2013), preprint |
| [39] | Zhang, C., Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type, Can. J. Math., 70, 2, 451-480, (2018) · Zbl 1423.14180 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
