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Arithmetic models for Hilbert modular varieties. (English) Zbl 0890.14010
Let \(F\) be a totally real field of degree \(d\) over \(\mathbb{Q}\), and discriminant \(\Delta\) with ring of integers \(\mathfrak O\) and \(\mathfrak A\) a square-free ideal of \(\mathfrak O\). The group \(SL(2,{\mathfrak O})\) acts on the product of \(d\) copies of the upper half plane \(\mathcal H\) via the \(d\) embeddings of \(SL(2,{\mathfrak O})\) into \(SL(2,\mathbb{R})\) induced by the embeddings of \(F\) into \(\mathbb{R}\). Let \(n\) be an integer relatively prime to \(\mathcal A\) and let \(\Gamma(n)\) denote the kernel of the reduction \(SL(2,{\mathfrak O})\to SL(2,{\mathfrak O}/n{\mathfrak O})\). The modular varieties of the title are \(H_\Gamma={\mathcal H}^d/\Gamma\) where \(\Gamma\) is either \[ \Gamma_0({\mathfrak A},n)=\Gamma_0({\mathfrak A})\cap \Gamma(n) \] or \[ \Gamma_{00}({\mathfrak A},n)=\Gamma_{00}({\mathfrak A})\cap \Gamma(n) \] the subgroups \(\Gamma_0({\mathfrak A}),\Gamma_{00}({\mathfrak A})\) of \(SL(2,{\mathfrak O})\) being defined by \[ \begin{aligned} & \Gamma_0({\mathcal A})=\left\{{a\;b\choose c\;d}\Bigl|c\in {\mathcal A}\right\}\\ & \Gamma_{00}({\mathcal A})=\left\{{a\;b\choose c\;d}\Bigl|c,a-1, d-1\in {\mathcal A}\right\}.\end{aligned} \] The varieties are defined over \(\mathbb{Q}(\zeta_n)\), \(\zeta_n=e^{2\pi i/n}\). The author studies models for \(H_\Gamma\) over \(\text{Spec }\mathbb{Z}[\zeta_n,\frac{1}{n}]\) and in particular the local structure of the reduction at primes, \(p\), \((p,n)=1\), and \(p\mid \text{Norm}({\mathcal A})\). Using the method of integral models due to M. Rapoport [Compos. Math. 36, 255-335 (1978; Zbl 0386.14006)], the author describes models for \(H_\Gamma\) that are smooth over \(\mathbb{Z}[\zeta_n,1/(n.\Delta.\text{Norm}({\mathfrak A}))]\), but difficulties arise in the case of models over \(\mathbb{Z}[\zeta_n,\frac1n ]\). In that case when the level of the subgroup is not invertible in the base scheme, it is not clear what the moduli problem should be.
The author sets out to understand the nature of the proper models by considering a notion of \(\mathfrak A\)-level structure for abelian varieties over the primes \(p\), \(p\mid\text{Norm}({\mathfrak A})\). The main result is that the moduli spaces are normal and relative complete intersections and the singularities are the same for Shimura varieties associated to forms of the reductive group. – The author also obtains a construction of regular scheme models for certain Hilbert-Blumenthal surfaces of the form \(H_{\Gamma_{00}({\mathcal A},1)}\) over \(\text{Spec }\mathbb{Z}\) with no prime inverted. Some of the problems that arise in the cases of non-square-free level are mentioned, but not dealt with.
The paper uses the language of algebraic stacks [see P. Deligne and D. Mumford, Publ. Math., Inst. Hautes Étud. Sci. 36, 75-109 (1969; Zbl 0181.48803)], but the reader who is unfamiliar with the language (as was the reviewer until he read this paper and that one – whence the unconscionable delay in the review itself) can assume that \(n\geq 3\), in which case the results refer to algebraic spaces or schemes, but the effort to understand the background is well worth while.

MSC:
14G35 Modular and Shimura varieties
14K10 Algebraic moduli of abelian varieties, classification
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References:
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