Integral canonical models of unitary Shimura varieties. (English) Zbl 1215.11061

Vasiu, Adrian

Asian J. Math. 12, No. 2, 151-176 (2008).

One associates Shimura varieties with a Shimura datum \((G, X)\) (where \(G\) is a reductive group over \(\mathbb{Q}\), \(X\) is a \(G(\mathbb{R})\)-conjugacy class of homomorphisms \(\mathrm{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m \rightarrow G_{\mathbb{R}}\), and the pair satisfies Deligne’s axioms) by taking the double quotients \(Sh(G,X)_K = G(\mathbb{Q}) \backslash X \times G(\mathbb{A}_f) / K\), where \(K\) ranges over compact open subgroups of the finite adelic points \(G(\mathbb{A}_f)\). The term “unitary” in the title means that the adjoint group \(G^{\mathrm{ad}}\) is nontrivial and its simple factors over \(\overline{\mathbb{Q}}\) are all of type \(A\) as simple algebraic groups.
The Shimura varieties are quasi-projective complex algebraic varieties by the Baily-Borel theorem, and by Shimura, Deligne, Milne and others, one knows that they admit canonical models over a number field \(E(G, X)\), the reflex field.
In order to study the arithmetic of these varieties, it is important to have good integral models. When the level subgroup \(K\) is hyperspecial at \(p\), Langlands suggested the existence of such models over \(\mathbb{Z}_p\), and Milne formulated the notion of integral canonical models, in terms of Néron-like extension properties, and conjectured their existence.
The main goal of this paper is to prove Milne’s conjecture in the unitary case. A very rough outline of the rather intricate proof is: first find another Shimura datum \((G_1, X_1)\) which has the same adjoint datum as \((G, X)\) and, in addition, embeds nicely into the Siegel modular case; prove the conjecture for \((G_1, X_1)\) using the embedding; and then deduce the same for \((G, X)\).
The article includes errata to the author’s previous article [Asian. J. Math., 3, 401–518 (1999; Zbl 1002.11052)] at the end.

Reviewer: Junecue Suh (Princeton)

MSC:

11G18	Arithmetic aspects of modular and Shimura varieties
14F30	\(p\)-adic cohomology, crystalline cohomology
14G35	Modular and Shimura varieties
14K10	Algebraic moduli of abelian varieties, classification
11G10	Abelian varieties of dimension \(> 1\)
14G40	Arithmetic varieties and schemes; Arakelov theory; heights

Keywords:

Shimura varieties; integral canonical models

Citations:

Zbl 1002.11052