Foliations on Shimura varieties in positive characteristic
Authors:
Eyal Z. Goren and Ehud de Shalit
Journal:
J. Algebraic Geom. 33 (2024), 401-454
DOI:
https://doi.org/10.1090/jag/820
Published electronically:
September 5, 2023
Full-text PDF
Abstract |
References |
Additional Information
Abstract: This paper is a continuation of a paper by de Shalit and Goren from 2018. We study foliations of two types on Shimura varieties $S$ in characteristic $p$. The first, which we call tautological foliations, are defined on Hilbert modular varieties, and lift to characteristic $0$. The second, the $V$-foliations, are defined on unitary Shimura varieties in characteristic $p$ only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are $p$-closed, and the locus where they are smooth. Where not smooth, we construct a successive blowup of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of $S$ by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at $p$, to $S.$
References
- F. Andreatta and E. Z. Goren, Hilbert modular forms: mod $p$ and $p$-adic aspects, Mem. Amer. Math. Soc. 173 (2005), no. 819, vi+100. MR 2110225, DOI 10.1090/memo/0819
- Jean-Benoît Bost, Algebraic leaves of algebraic foliations over number fields, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 161–221 (English, with English and French summaries). MR 1863738, DOI 10.1007/s10240-001-8191-3
- Marco Brunella, Birational geometry of foliations, IMPA Monographs, vol. 1, Springer, Cham, 2015. MR 3328860, DOI 10.1007/978-3-319-14310-1
- Fred Diamond and Payman L. Kassaei, Minimal weights of Hilbert modular forms in characteristic $p$, Compos. Math. 153 (2017), no. 9, 1769–1778. MR 3668413, DOI 10.1112/S0010437X17007230
- Torsten Ekedahl, Foliations and inseparable morphisms, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 139–149. MR 927978, DOI 10.1090/pspum/046.2/927978
- T. Ekedahl, N. I. Shepherd-Barron, and R. L. Taylor, A conjecture on the existence of compact leaves of algebraic foliations, Preprint.
- Eyal Z. Goren, Hasse invariants for Hilbert modular varieties, Israel J. Math. 122 (2001), 157–174. MR 1826497, DOI 10.1007/BF02809897
- Ehud de Shalit and Eyal Z. Goren, Foliations on unitary Shimura varieties in positive characteristic, Compos. Math. 154 (2018), no. 11, 2267–2304. MR 3866508, DOI 10.1112/s0010437x18007406
- Eyal Z. Goren and Payman L. Kassaei, Canonical subgroups over Hilbert modular varieties, J. Reine Angew. Math. 670 (2012), 1–63. MR 2982691, DOI 10.1515/CRELLE.2011.149
- E. Z. Goren and F. Oort, Stratifications of Hilbert modular varieties, J. Algebraic Geom. 9 (2000), no. 1, 111–154. MR 1713522
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157, DOI 10.1007/978-1-4757-3849-0
- N. Jacobson, Basic algebra II, Dover Books in Mathematics, 2009.
- Nicholas M. Katz, $p$-adic $L$-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297. MR 513095, DOI 10.1007/BF01390187
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177, DOI 10.1007/BF02684688
- Tetsuzo Kimura and Hiroshi Niitsuma, On Kunz’s conjecture, J. Math. Soc. Japan 34 (1982), no. 2, 371–378. MR 651278, DOI 10.2969/jmsj/03420371
- Kai-Wen Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Mathematical Society Monographs Series, vol. 36, Princeton University Press, Princeton, NJ, 2013. MR 3186092, DOI 10.1515/9781400846016
- G. Livneh, Purely inseparable Galois morphisms and smooth foliations, M.Sc. Thesis, Hebrew University, 2021.
- Yoichi Miyaoka and Thomas Peternell, Geometry of higher-dimensional algebraic varieties, DMV Seminar, vol. 26, Birkhäuser Verlag, Basel, 1997. MR 1468476, DOI 10.1007/978-3-0348-8893-6
- Ben Moonen, Serre-Tate theory for moduli spaces of PEL type, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 2, 223–269 (English, with English and French summaries). MR 2061781, DOI 10.1016/j.ansens.2003.04.004
- Ben Moonen, Group schemes with additional structures and Weyl group cosets, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 255–298. MR 1827024
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 282985
- Georgios Pappas, Arithmetic models for Hilbert modular varieties, Compositio Math. 98 (1995), no. 1, 43–76. MR 1353285
- M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Math. 36 (1978), no. 3, 255–335 (French). MR 515050
- Erwan Rousseau and Frédéric Touzet, Curves in Hilbert modular varieties, Asian J. Math. 22 (2018), no. 4, 673–689. MR 3862056, DOI 10.4310/AJM.2018.v22.n4.a4
- Hellmuth Stamm, On the reduction of the Hilbert-Blumenthal-moduli scheme with $\Gamma _0(p)$-level structure, Forum Math. 9 (1997), no. 4, 405–455. MR 1457134, DOI 10.1515/form.1997.9.405
- The Stacks Project, https://stacks.math.columbia.edu.
- Eva Viehmann and Torsten Wedhorn, Ekedahl-Oort and Newton strata for Shimura varieties of PEL type, Math. Ann. 356 (2013), no. 4, 1493–1550. MR 3072810, DOI 10.1007/s00208-012-0892-z
- Torsten Wedhorn, Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 5, 575–618 (English, with English and French summaries). MR 1710754, DOI 10.1016/S0012-9593(01)80001-X
- Torsten Wedhorn, The dimension of Oort strata of Shimura varieties of PEL-type, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 441–471. MR 1827029
- Shuen Yuan, Inseparable Galois theory of exponent one, Trans. Amer. Math. Soc. 149 (1970), 163–170. MR 257063, DOI 10.1090/S0002-9947-1970-0257063-1
References
- F. Andreatta and E. Z. Goren, Hilbert modular forms: mod $p$ and $p$-adic aspects, Mem. Amer. Math. Soc. 173 (2005), no. 819, vi+100. MR 2110225, DOI 10.1090/memo/0819
- Jean-Benoît Bost, Algebraic leaves of algebraic foliations over number fields, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 161–221 (English, with English and French summaries). MR 1863738, DOI 10.1007/s10240-001-8191-3
- Marco Brunella, Birational geometry of foliations, IMPA Monographs, vol. 1, Springer, Cham, 2015. MR 3328860, DOI 10.1007/978-3-319-14310-1
- Fred Diamond and Payman L. Kassaei, Minimal weights of Hilbert modular forms in characteristic $p$, Compos. Math. 153 (2017), no. 9, 1769–1778. MR 3668413, DOI 10.1112/S0010437X17007230
- Torsten Ekedahl, Foliations and inseparable morphisms, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 139–149. MR 927978, DOI 10.1090/pspum/046.2/927978
- T. Ekedahl, N. I. Shepherd-Barron, and R. L. Taylor, A conjecture on the existence of compact leaves of algebraic foliations, Preprint.
- Eyal Z. Goren, Hasse invariants for Hilbert modular varieties, Israel J. Math. 122 (2001), 157–174. MR 1826497, DOI 10.1007/BF02809897
- Ehud de Shalit and Eyal Z. Goren, Foliations on unitary Shimura varieties in positive characteristic, Compos. Math. 154 (2018), no. 11, 2267–2304. MR 3866508, DOI 10.1112/s0010437x18007406
- Eyal Z. Goren and Payman L. Kassaei, Canonical subgroups over Hilbert modular varieties, J. Reine Angew. Math. 670 (2012), 1–63. MR 2982691, DOI 10.1515/CRELLE.2011.149
- E. Z. Goren and F. Oort, Stratifications of Hilbert modular varieties, J. Algebraic Geom. 9 (2000), no. 1, 111–154. MR 1713522
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- N. Jacobson, Basic algebra II, Dover Books in Mathematics, 2009.
- Nicholas M. Katz, $p$-adic $L$-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297. MR 513095, DOI 10.1007/BF01390187
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177
- Tetsuzo Kimura and Hiroshi Niitsuma, On Kunz’s conjecture, J. Math. Soc. Japan 34 (1982), no. 2, 371–378. MR 651278, DOI 10.2969/jmsj/03420371
- Kai-Wen Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Mathematical Society Monographs Series, vol. 36, Princeton University Press, Princeton, NJ, 2013. MR 3186092, DOI 10.1515/9781400846016
- G. Livneh, Purely inseparable Galois morphisms and smooth foliations, M.Sc. Thesis, Hebrew University, 2021.
- Yoichi Miyaoka and Thomas Peternell, Geometry of higher-dimensional algebraic varieties, DMV Seminar, vol. 26, Birkhäuser Verlag, Basel, 1997. MR 1468476, DOI 10.1007/978-3-0348-8893-6
- Ben Moonen, Serre-Tate theory for moduli spaces of PEL type, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 2, 223–269 (English, with English and French summaries). MR 2061781, DOI 10.1016/j.ansens.2003.04.004
- Ben Moonen, Group schemes with additional structures and Weyl group cosets, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 255–298. MR 1827024
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- Georgios Pappas, Arithmetic models for Hilbert modular varieties, Compositio Math. 98 (1995), no. 1, 43–76. MR 1353285
- M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Math. 36 (1978), no. 3, 255–335 (French). MR 515050
- Erwan Rousseau and Frédéric Touzet, Curves in Hilbert modular varieties, Asian J. Math. 22 (2018), no. 4, 673–689. MR 3862056, DOI 10.4310/AJM.2018.v22.n4.a4
- Hellmuth Stamm, On the reduction of the Hilbert-Blumenthal-moduli scheme with $\Gamma _0(p)$-level structure, Forum Math. 9 (1997), no. 4, 405–455. MR 1457134, DOI 10.1515/form.1997.9.405
- The Stacks Project, https://stacks.math.columbia.edu.
- Eva Viehmann and Torsten Wedhorn, Ekedahl-Oort and Newton strata for Shimura varieties of PEL type, Math. Ann. 356 (2013), no. 4, 1493–1550. MR 3072810, DOI 10.1007/s00208-012-0892-z
- Torsten Wedhorn, Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 5, 575–618 (English, with English and French summaries). MR 1710754, DOI 10.1016/S0012-9593(01)80001-X
- Torsten Wedhorn, The dimension of Oort strata of Shimura varieties of PEL-type, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 441–471. MR 1827029
- Shuen Yuan, Inseparable Galois theory of exponent one, Trans. Amer. Math. Soc. 149 (1970), 163–170. MR 257063, DOI 10.2307/1995668
Additional Information
Eyal Z. Goren
Affiliation:
McGill University, Montréal, Québec, Canada
MR Author ID:
623278
Email:
eyal.goren@mcgill.ca
Ehud de Shalit
Affiliation:
Hebrew University of Jerusalem, Israel
MR Author ID:
221112
Email:
ehud.deshalit@mail.huji.ac.il
Received by editor(s):
May 2, 2022
Received by editor(s) in revised form:
May 22, 2022, and January 1, 2023
Published electronically:
September 5, 2023
Additional Notes:
The authors’ research was supported by ISF grant 276.17 and NSERC grant 223148.
Article copyright:
© Copyright 2023
University Press, Inc.
