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The $$\mu$$-ordinary Hasse invariant of unitary Shimura varieties. (English) Zbl 1391.14045
Let $$A$$ 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.

##### MSC:
 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties
##### Keywords:
Shimura variety; Hasse invariant; abelian variety; unitary group
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##### References:
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