## 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.

### 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

Zbl 0346.14010
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