##
**Period spaces for \(p\)-divisible groups.**
*(English)*
Zbl 0873.14039

Annals of Mathematics Studies. 141. Princeton, NJ: Princeton Univ. Press. xxi, 324 p., $ 59.50; £50.00/hbk (1996).

Let \(E\) be a \(p\)-adic field, and let \(\Omega^d_E\) be the completion of all \(E\)-rational hyperplanes in the projective space \(\mathbb P^d\). This is a rigid-analytic space over \(E\) equipped with an action of \(\text{GL}_d(E)\). V. G. Drinfel’d [Funct. Anal. Appl. 10, 107-115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29-40 (1976; Zbl 0346.14010)] has constructed a system of unramified coverings \(\widetilde \Omega^d_E\) of \(\Omega^d_E\) to which the action of \(\text{GL}_d(E)\) is lifted. Drinfeld has shown that these covering spaces can be used to \(p\)-adically uniformize the rigid-analytic spaces corresponding to Shimura varieties associated to certain unitary groups. Also Drinfeld has conjectured that the \(\ell\)-adic cohomology group with compact supports \(H_e^i (\widetilde \Omega^d_E \otimes E,\overline {\mathbb Q}_\ell)\) \((\ell\neq p)\) should give a realization of all supercuspidal representations of \(\text{GL}_d(E)\).

In this monograph, the authors generalize Drinfeld’s construction and results to other \(p\)-adic groups. Their construction is based on the moduli theory of \(p\)-divisible groups of a fixed isogeny type. The moduli spaces constructed here are formal schemes over the ring of integers \({O}_E\) whose generic fibers yield rigid-analytic spaces generalizing Drinfeld’s \(\Omega^d_E\), and the covering spaces are obtained by trivializing the Tate modules of the universal \(p\)-divisible groups over these formal schemes. Moreover, the authors show that these spaces may be used to uniformize the rigid-analytic spaces associated to general Shimura varieties. Also the authors exhibit a rigid-analytic period map from the covering spaces to one of the \(p\)-adic symmetric spaces associated to the \(p\)-adic group. – The main results are presented in the following manner: First the moduli problems of \(p\)-divisible groups and the representability theorem (which yields the formal schemes generalizing \(\widetilde \Omega^d_E)\) are described. Then the covering spaces and the rigid-analytic period morphisms are described. Finally the non-archimedean uniformization theorem for Shimura varieties is proved. The main results are now described in more detail.

The moduli problem of \(p\)-divisible groups is divided into two types:

(EL): this type parametrizes \(p\)-divisible groups with endomorphisms and level structures within a fixed isogeny class;

(PEL): this type parametrizes \(p\)-divisible groups with polarizations, endomorphisms and level structures within a fixed isogeny class.

Let \(p\) be a prime \((\neq 2\) for most results). Let \(L\) be an algebraically closed field of characteristic \(p\), \(W(L)\) be the ring of Witt vectors over \(L\), and \(K_0= K_0(L)= W(L) \otimes_\mathbb Z \mathbb Q\), and let \(\sigma\) be the Frobenius automorphism of \(K_0\). Denote by \(\text{Nilp}_{W(L)}\) the category of locally Noetherian schemes over \(\text{Spec} W(L)\) such that the ideal sheaf \(p\cdot {\mathcal O}_{W(L)}\) is locally nilpotent. Let \({\mathfrak X}\) be a \(p\)-divisible group over \(\text{Spec} L\). Let \({\mathcal M}\) be a functor, which associates to \(S\in \text{Nilp} W(L)\) the set of isomorphism classes of pairs \((X,\rho)\) consisting of a \(p\)-divisible group \(X\) over \(S\) and a quasi-isogeny \(\rho: {\mathfrak X} \times_{\text{Spec} L} S\to X\times_S \overline S\) of \(p\)-divisible groups over \(\overline S\). Then \({\mathcal M}\) is representable by a formal scheme locally formally of finite type over \(\text{Spf} W(L)\). This result allows one to establish the representability of the functors (denoted \(\breve {\mathcal M})\) of \(p\)-divisible groups endowed with endomorphisms and level structures (EL), respectively, with polarizations and endomorphisms and level structures (PEL), by formal schemes locally formally of finite type over \(\text{Spf} {O}_{\breve E}\). (Here \(\breve E=E \cdot K_0\) with \({O}_{\breve E}\) the ring of integers.) – These functors are shown to depend on certain “rational” and “integral” data. For instance, the “rational” data of type (EL) consists of a 4-tuple \((B,V,b,\mu)\), where \(B\) is a finite dimensional semi-simple \(\mathbb Q_p\)-algebra, \(V\) a finite left \(B\)-module. Let \(G=GL_B(V)\) (algebraic group over \(\mathbb Q_p)\). Let \(b\in G(K_0)\), and \(\mu: \mathbb G_m \to G_K\) a homomorphism defined over a finite extension \(K/K_0\). One requires that the filtered isocrystal over \(K\), \((V \otimes_{\mathbb Q_p} K_0\), \(b(\text{id} \otimes \sigma)\), \(V^\bullet_K)\), is that of associated to a \(p\)-divisible group over \(\text{Spec} {O}_K\). For each pair \((G,b)\) as above, there is the group \(J(\mathbb Q_p)\) of quasi-isogenies of \({\mathfrak X}\). “Integral” data of type (EL) consists of a maximal order \({O}_B\) and an \({O}_B\)-lattice chain \({\mathcal L}\) in \(V\). Similarly “rational” and “integral” data of type (PEL) are defined. – Several examples of the formal schemes \(\breve {\mathcal M}\) are given.

Next, the period morphism associated to the moduli problem \(\breve {\mathcal M}\) of (EL) or (PEL)-type is described. Let \(B,V\) and \(G\) be as above. Let \(\breve {\mathcal M}^{\text{rig}}\) be the rigid-analytic space over \(\breve E\) associated to \(\breve {\mathcal M}\) (the generic fibre of \(\breve {\mathcal M})\). Then the period morphism is defined as a rigid-analytic morphism from \(\breve {\mathcal M}\) to \(\breve {\mathcal F}^{\text{rig}}= {\mathcal F} \times_{\text{Spec} E} \breve E\), where \({\mathcal F}\) is the homogeneous projective algebraic variety under \(G_{\breve E}\) defined by the conjugacy class of the one-parameter subgroup \(\mu\). – Further, properties of the period morphisms are discussed, e.g., \(\breve \pi\) is étale and \(J(\mathbb Q_p)\)-equivariant, and the descriptions of the image of \(\breve\pi\) and the associated Tate modules.

Finally, a non-archimedean uniformisation theorem for certain Shimura varieties is proved. Let \(B\) stand for a finite dimensional algebra over \(\mathbb Q\) equipped with a positive anti-involution *, \(V\) a finite left \(B\)-module with a non-degenerate alternating bilinear form \(( , )\) with values in \(\mathbb Q\) satisfying certain conditions and \(G=\text{GL}_B(V)\) (algebraic group over \(\mathbb Q)\). Then one obtains a Shimura variety over \(E\subset \mathbb C\) associated to \((G,h: \text{Res}_{\mathbb C/ \mathbb R} \mathbb G_m \to G_\mathbb R)\). Fix data of type (PEL), i.e. \((B \otimes \mathbb Q_p,^*,V \otimes \mathbb Q_p\), \(( , ),b,\mu, {O}_B \otimes \mathbb Z_p, \Lambda)\). Here an order \({O}_B\) of \(B\) is chosen so that \({O}_B \otimes \mathbb Z_p\) is a maximal order of \(B \otimes_\mathbb Q \mathbb Q_p\) stable under *, and a self-dual \({O}_B \otimes_\mathbb Z\) \(\mathbb Z_p\)-lattice \(\Lambda\) in \(V \otimes_\mathbb Q \mathbb Q_p\). Fix an open compact subgroup \(C^p \subset G(\mathbb A^p_f)\). These data define a moduli problem of (PEL) type parametrizing triples \((A,\overline \lambda, \overline \eta^p)\) consisting of an \({O}_B\)-abelian variety \(A\), a \(\mathbb Q\)-homogeneous principal \({O}_B\)-polarization \(\overline \lambda\), and a \(C^p\)-level structure \(\overline \eta^p\) and which is representable by a quasi-projective scheme \({\mathcal A}_{C^p}\) over \(\text{Spec} {O}_{E_\nu}\). The generic fiber of \({\mathcal A}_{C^p}\) contains the Shimura variety \((G,h)\) as an open and closed subscheme. Now fix a point \((A_0, \overline \lambda_0, \overline \eta^p_0)\) of \({\mathcal A}_{C^p} (L)\) and assume that it is basic. Then the set of points \((A,\overline \lambda, \overline \eta^p)\) of \({\mathcal A}_{C^p} (L)\) such that \((A, \overline \lambda)\) is isogenous to \((A_0, \overline \lambda_0)\) is a closed subset \(Z\) of \({\mathcal A}_{C^p}\). If \({\mathcal A}_{C^p} |Z\) denotes the formal completion of \({\mathcal A}_{C^p}\) along \(Z\), then there is an isomorphism of formal schemes over \(\text{Spf} {O}_{E_\nu}: I(\mathbb Q) \backslash [{\mathcal M} \times G (\mathbb A^p_f)/C^p] \sim {\mathcal A}_{C^p} |Z\). (Here \(I(\mathbb Q)\) is the group of quasi-isogenies of \((A_0, \overline \lambda_0)\) that acts diagonally through suitable embeddings \(I(\mathbb Q) \to J (\mathbb Q_p)\); \(I(\mathbb Q) \to G(\mathbb A^p_f)\).)

The contents of this monograph is as follows: \(\S 1\): \(p\)-adic symmetric domains, \(\S 2\): Quasi-isogenies of \(p\)-divisible groups, \(\S 3\): Moduli spaces of \(p\)-divisible groups (with Appendix: Normal forms of lattice chains), \(\S 4\): The formal Hecke correspondence, \(\S 5\): The period morphism and the rigid-analytic coverings, \(\S 6\): The \(p\)-adic uniformization of Shimura varieties, and bibliography and index.

In this monograph, the authors generalize Drinfeld’s construction and results to other \(p\)-adic groups. Their construction is based on the moduli theory of \(p\)-divisible groups of a fixed isogeny type. The moduli spaces constructed here are formal schemes over the ring of integers \({O}_E\) whose generic fibers yield rigid-analytic spaces generalizing Drinfeld’s \(\Omega^d_E\), and the covering spaces are obtained by trivializing the Tate modules of the universal \(p\)-divisible groups over these formal schemes. Moreover, the authors show that these spaces may be used to uniformize the rigid-analytic spaces associated to general Shimura varieties. Also the authors exhibit a rigid-analytic period map from the covering spaces to one of the \(p\)-adic symmetric spaces associated to the \(p\)-adic group. – The main results are presented in the following manner: First the moduli problems of \(p\)-divisible groups and the representability theorem (which yields the formal schemes generalizing \(\widetilde \Omega^d_E)\) are described. Then the covering spaces and the rigid-analytic period morphisms are described. Finally the non-archimedean uniformization theorem for Shimura varieties is proved. The main results are now described in more detail.

The moduli problem of \(p\)-divisible groups is divided into two types:

(EL): this type parametrizes \(p\)-divisible groups with endomorphisms and level structures within a fixed isogeny class;

(PEL): this type parametrizes \(p\)-divisible groups with polarizations, endomorphisms and level structures within a fixed isogeny class.

Let \(p\) be a prime \((\neq 2\) for most results). Let \(L\) be an algebraically closed field of characteristic \(p\), \(W(L)\) be the ring of Witt vectors over \(L\), and \(K_0= K_0(L)= W(L) \otimes_\mathbb Z \mathbb Q\), and let \(\sigma\) be the Frobenius automorphism of \(K_0\). Denote by \(\text{Nilp}_{W(L)}\) the category of locally Noetherian schemes over \(\text{Spec} W(L)\) such that the ideal sheaf \(p\cdot {\mathcal O}_{W(L)}\) is locally nilpotent. Let \({\mathfrak X}\) be a \(p\)-divisible group over \(\text{Spec} L\). Let \({\mathcal M}\) be a functor, which associates to \(S\in \text{Nilp} W(L)\) the set of isomorphism classes of pairs \((X,\rho)\) consisting of a \(p\)-divisible group \(X\) over \(S\) and a quasi-isogeny \(\rho: {\mathfrak X} \times_{\text{Spec} L} S\to X\times_S \overline S\) of \(p\)-divisible groups over \(\overline S\). Then \({\mathcal M}\) is representable by a formal scheme locally formally of finite type over \(\text{Spf} W(L)\). This result allows one to establish the representability of the functors (denoted \(\breve {\mathcal M})\) of \(p\)-divisible groups endowed with endomorphisms and level structures (EL), respectively, with polarizations and endomorphisms and level structures (PEL), by formal schemes locally formally of finite type over \(\text{Spf} {O}_{\breve E}\). (Here \(\breve E=E \cdot K_0\) with \({O}_{\breve E}\) the ring of integers.) – These functors are shown to depend on certain “rational” and “integral” data. For instance, the “rational” data of type (EL) consists of a 4-tuple \((B,V,b,\mu)\), where \(B\) is a finite dimensional semi-simple \(\mathbb Q_p\)-algebra, \(V\) a finite left \(B\)-module. Let \(G=GL_B(V)\) (algebraic group over \(\mathbb Q_p)\). Let \(b\in G(K_0)\), and \(\mu: \mathbb G_m \to G_K\) a homomorphism defined over a finite extension \(K/K_0\). One requires that the filtered isocrystal over \(K\), \((V \otimes_{\mathbb Q_p} K_0\), \(b(\text{id} \otimes \sigma)\), \(V^\bullet_K)\), is that of associated to a \(p\)-divisible group over \(\text{Spec} {O}_K\). For each pair \((G,b)\) as above, there is the group \(J(\mathbb Q_p)\) of quasi-isogenies of \({\mathfrak X}\). “Integral” data of type (EL) consists of a maximal order \({O}_B\) and an \({O}_B\)-lattice chain \({\mathcal L}\) in \(V\). Similarly “rational” and “integral” data of type (PEL) are defined. – Several examples of the formal schemes \(\breve {\mathcal M}\) are given.

Next, the period morphism associated to the moduli problem \(\breve {\mathcal M}\) of (EL) or (PEL)-type is described. Let \(B,V\) and \(G\) be as above. Let \(\breve {\mathcal M}^{\text{rig}}\) be the rigid-analytic space over \(\breve E\) associated to \(\breve {\mathcal M}\) (the generic fibre of \(\breve {\mathcal M})\). Then the period morphism is defined as a rigid-analytic morphism from \(\breve {\mathcal M}\) to \(\breve {\mathcal F}^{\text{rig}}= {\mathcal F} \times_{\text{Spec} E} \breve E\), where \({\mathcal F}\) is the homogeneous projective algebraic variety under \(G_{\breve E}\) defined by the conjugacy class of the one-parameter subgroup \(\mu\). – Further, properties of the period morphisms are discussed, e.g., \(\breve \pi\) is étale and \(J(\mathbb Q_p)\)-equivariant, and the descriptions of the image of \(\breve\pi\) and the associated Tate modules.

Finally, a non-archimedean uniformisation theorem for certain Shimura varieties is proved. Let \(B\) stand for a finite dimensional algebra over \(\mathbb Q\) equipped with a positive anti-involution *, \(V\) a finite left \(B\)-module with a non-degenerate alternating bilinear form \(( , )\) with values in \(\mathbb Q\) satisfying certain conditions and \(G=\text{GL}_B(V)\) (algebraic group over \(\mathbb Q)\). Then one obtains a Shimura variety over \(E\subset \mathbb C\) associated to \((G,h: \text{Res}_{\mathbb C/ \mathbb R} \mathbb G_m \to G_\mathbb R)\). Fix data of type (PEL), i.e. \((B \otimes \mathbb Q_p,^*,V \otimes \mathbb Q_p\), \(( , ),b,\mu, {O}_B \otimes \mathbb Z_p, \Lambda)\). Here an order \({O}_B\) of \(B\) is chosen so that \({O}_B \otimes \mathbb Z_p\) is a maximal order of \(B \otimes_\mathbb Q \mathbb Q_p\) stable under *, and a self-dual \({O}_B \otimes_\mathbb Z\) \(\mathbb Z_p\)-lattice \(\Lambda\) in \(V \otimes_\mathbb Q \mathbb Q_p\). Fix an open compact subgroup \(C^p \subset G(\mathbb A^p_f)\). These data define a moduli problem of (PEL) type parametrizing triples \((A,\overline \lambda, \overline \eta^p)\) consisting of an \({O}_B\)-abelian variety \(A\), a \(\mathbb Q\)-homogeneous principal \({O}_B\)-polarization \(\overline \lambda\), and a \(C^p\)-level structure \(\overline \eta^p\) and which is representable by a quasi-projective scheme \({\mathcal A}_{C^p}\) over \(\text{Spec} {O}_{E_\nu}\). The generic fiber of \({\mathcal A}_{C^p}\) contains the Shimura variety \((G,h)\) as an open and closed subscheme. Now fix a point \((A_0, \overline \lambda_0, \overline \eta^p_0)\) of \({\mathcal A}_{C^p} (L)\) and assume that it is basic. Then the set of points \((A,\overline \lambda, \overline \eta^p)\) of \({\mathcal A}_{C^p} (L)\) such that \((A, \overline \lambda)\) is isogenous to \((A_0, \overline \lambda_0)\) is a closed subset \(Z\) of \({\mathcal A}_{C^p}\). If \({\mathcal A}_{C^p} |Z\) denotes the formal completion of \({\mathcal A}_{C^p}\) along \(Z\), then there is an isomorphism of formal schemes over \(\text{Spf} {O}_{E_\nu}: I(\mathbb Q) \backslash [{\mathcal M} \times G (\mathbb A^p_f)/C^p] \sim {\mathcal A}_{C^p} |Z\). (Here \(I(\mathbb Q)\) is the group of quasi-isogenies of \((A_0, \overline \lambda_0)\) that acts diagonally through suitable embeddings \(I(\mathbb Q) \to J (\mathbb Q_p)\); \(I(\mathbb Q) \to G(\mathbb A^p_f)\).)

The contents of this monograph is as follows: \(\S 1\): \(p\)-adic symmetric domains, \(\S 2\): Quasi-isogenies of \(p\)-divisible groups, \(\S 3\): Moduli spaces of \(p\)-divisible groups (with Appendix: Normal forms of lattice chains), \(\S 4\): The formal Hecke correspondence, \(\S 5\): The period morphism and the rigid-analytic coverings, \(\S 6\): The \(p\)-adic uniformization of Shimura varieties, and bibliography and index.

Reviewer: Noriko Yui (Kingston)

### MSC:

14L05 | Formal groups, \(p\)-divisible groups |

14G20 | Local ground fields in algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

32G20 | Period matrices, variation of Hodge structure; degenerations |