The \(\mu\)-ordinary Hasse invariant of unitary Shimura varieties.

*(English)*Zbl 1391.14045Let \(A\) be an abelian scheme over an \(\mathbb{F}_p\)-scheme \(S\) with identity section \(e_A: S \to A\). Let \(\omega_A := \mathrm{det} (e_A^*\Omega_{A/S})\) denote the Hodge line bundle on \(S\). The Verschiebung isogeny \(V_{A/S}: A^{(p)} \to A\) induces a pullback morphism \(V^*_{A/S} \Omega_{A/S} \to \Omega_{A^{(p)}/S}\) on sheaves of relative \(1\)-forms, and hence, applying \(\mathrm{det} \circ e_{A^{(p)}}^*\), a morphism \(\omega_A \to \omega_{A^{(p)}} \cong \omega_A^{\otimes p}\). This defines a canonical global section \(H \in \mathrm{H}^0(S,\omega_A^{\otimes p-1})\), called the Hasse invariant of \(A\). The ordinary locus in \(S\) is the locus of points \(x \in S\) over which the fiber \(A_x\) is an ordinary abelian variety, i.e., \(A_x[p](\overline{\kappa(x)})\) has the maximum possible number \(p^{\dim }A_x\) of points for an abelian variety in characteristic \(p\). The ordinary locus may then be described as the nonvanishing locus of \(H\), since it is easy to see that both loci are the set of points \(x\) such that the Verschiebung of \(A_x\) is separable.

The Hasse invariant is plainly defined over the moduli stack of abelian varieties \(\mathcal{M}\) over \(\mathbb{F}_p\), and hence (via pullback) over any space mapping to \(\mathcal{M}\). In particular, one may speak of the Hasse invariant and the ordinary locus in the fiber over a prime \(\mathfrak{p}\) of an integral model of a Shimura variety of PEL type, where \(\mathfrak{p}\) is a prime over \(p\) in the reflex field \(E\) of the Shimura variety. In this case, when the level subgroup is hyperspecial at \(p\), T. Wedhorn [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 5, 575–618 (1999; Zbl 0983.14024)] showed that the ordinary locus is nonempty (in which case it is dense in the fiber) if and only if \(\mathbb{Q}_p = E_{\mathfrak{p}}\) – a rather restrictive condition. Wedhorn furthermore showed that a better general notion in this context is that of so-called \(\mu\)-ordinariness (defined in a group-theoretic way in terms of the Shimura datum), insofar as the \(\mu\)-ordinary locus is always open and dense in the fiber.

It is then natural to ask whether the \(\mu\)-ordinary locus can also be described as the nonvanishing locus of a section of a line bundle. In the paper under review, the authors show that this is so in the case of PEL data that give rise to a unitary similitude group (still with hyperspecial level at \(p\)). In fact, their “\(\mu\)-ordinary Hasse invariant” is again a section of an explicit power of the Hodge line bundle. The authors furthermore show that the \(\mu\)-ordinary Hasse invariant is compatible with varying the prime-to-\(p\) level, that a power of it extends to the minimal compactification of the fiber of the Shimura variety, and that a power of it lifts to characteristic zero; all of these properties are known to hold for the classical Hasse invariant \(H\). As applications, the authors deduce that the \(\mu\)-ordinary locus of the minimal compactification of the fiber is affine, as well as a generalization of the main result of previous work of the first author [Compos. Math. 150, No. 2, 191–228 (2014; Zbl 1385.11044)], which concerns attaching Galois representations to automorphic representations whose archimedean component is a holomorphic limit of discrete series. The main technical idea in the paper is to use the action of the absolute Frobenius on the crystalline cohomology of abelian varieties.

The Hasse invariant is plainly defined over the moduli stack of abelian varieties \(\mathcal{M}\) over \(\mathbb{F}_p\), and hence (via pullback) over any space mapping to \(\mathcal{M}\). In particular, one may speak of the Hasse invariant and the ordinary locus in the fiber over a prime \(\mathfrak{p}\) of an integral model of a Shimura variety of PEL type, where \(\mathfrak{p}\) is a prime over \(p\) in the reflex field \(E\) of the Shimura variety. In this case, when the level subgroup is hyperspecial at \(p\), T. Wedhorn [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 5, 575–618 (1999; Zbl 0983.14024)] showed that the ordinary locus is nonempty (in which case it is dense in the fiber) if and only if \(\mathbb{Q}_p = E_{\mathfrak{p}}\) – a rather restrictive condition. Wedhorn furthermore showed that a better general notion in this context is that of so-called \(\mu\)-ordinariness (defined in a group-theoretic way in terms of the Shimura datum), insofar as the \(\mu\)-ordinary locus is always open and dense in the fiber.

It is then natural to ask whether the \(\mu\)-ordinary locus can also be described as the nonvanishing locus of a section of a line bundle. In the paper under review, the authors show that this is so in the case of PEL data that give rise to a unitary similitude group (still with hyperspecial level at \(p\)). In fact, their “\(\mu\)-ordinary Hasse invariant” is again a section of an explicit power of the Hodge line bundle. The authors furthermore show that the \(\mu\)-ordinary Hasse invariant is compatible with varying the prime-to-\(p\) level, that a power of it extends to the minimal compactification of the fiber of the Shimura variety, and that a power of it lifts to characteristic zero; all of these properties are known to hold for the classical Hasse invariant \(H\). As applications, the authors deduce that the \(\mu\)-ordinary locus of the minimal compactification of the fiber is affine, as well as a generalization of the main result of previous work of the first author [Compos. Math. 150, No. 2, 191–228 (2014; Zbl 1385.11044)], which concerns attaching Galois representations to automorphic representations whose archimedean component is a holomorphic limit of discrete series. The main technical idea in the paper is to use the action of the absolute Frobenius on the crystalline cohomology of abelian varieties.

Reviewer: Brian Smithling (Baltimore)

##### References:

[1] | P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton 1978. · Zbl 0383.14010 |

[2] | M. Demazure, Lectures on p-divisible groups, Lecture Notes in Math. 302, Springer, Berlin 1972. · Zbl 0247.14010 |

[3] | A. J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. Inst. Hautes Études Sci. 86 (1995), 5-96. · Zbl 0864.14009 |

[4] | W. Goldring, Galois representations associated to holomorphic limits of discrete series. With an appendix by S.-W. Shin, Compos. Math. 150 (2014), 191-228. · Zbl 1309.11038 |

[5] | W. Goldring and M.-H. Nicole, A μ-ordinary Hasse invariant, preprint (2013), . |

[6] | W. Goldring and M.-H. Nicole, The μ-ordinary Hasse invariant of unitary Shimura varieties, preprint (2013), . |

[7] | M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Ann. of Math. Stud. 151, Princeton University Press, Princeton 2001. · Zbl 1036.11027 |

[8] | N. Katz, Slope filtration of {{F}}-crystals, Astérisque 63 (1979), 113-164. · Zbl 0426.14007 |

[9] | J.-S. Koskivirta and T. Wedhorn, Generalized Hasse invariants for Shimura varieties of Hodge type, preprint (2014), . |

[10] | K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Math. Soc. Monogr. Ser. 36, Princeton University Press, Princeton 2013. · Zbl 1284.14004 |

[11] | K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties, Duke Math. J. 161 (2012), no. 6, 1113-1170. · Zbl 1296.11072 |

[12] | K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties, Adv. Math. 242 (2013), 228-286. · Zbl 1276.11103 |

[13] | B. Moonen, Serre-Tate theory for moduli spaces of PEL-type, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), no. 2, 223-269.. · Zbl 1107.11028 |

[14] | D. Mumford, Abelian varieties, Tata Inst. Fund. Res. Stud. Math. 5, Hindustan Book Agency, New Delhi 2008. |

[15] | M. Rapoport and M. Richartz, On the classification and specialization of {{F}}-isocrystals with additional structure, Compos. Math. 103 (1996), 153-181. · Zbl 0874.14008 |

[16] | T. Wedhorn, Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. Éc. Norm. Supér. (4) 32 (1999), no. 5, 575-618. · Zbl 0983.14024 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.