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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.

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)

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\textit{W. Goldring} and \textit{M.-H. Nicole}, J. Reine Angew. Math. 728, 137--151 (2017; Zbl 1391.14045)

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