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Appendix to “The Iwasawa conjecture for totally real fields” by A.Wiles: Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces. (English) Zbl 0754.14030
[This paper is an appendix to A. Wiles, ibid. 493-540 (1990; Zbl 0719.11071)].
Let $$F$$ be a totally real number field, $$g=[F:\mathbb{Q}]>1$$, $${\mathfrak O}_ F$$ its ring of integers. Let $${\mathfrak c}^ \#$$ denote a projective rank-one $${\mathfrak O}_ F$$-module with positivity. Denote by $${\mathcal M}$$, respectively $${\mathcal M}({\mathfrak c}^ \#)$$, respectively $${\mathcal M}_ n({\mathfrak c}^ \#)$$ the moduli stack which classifies all HBAV ( Hilbert- Blumenthal abelian varieties) of relative dimension $$g$$, respectively all $${\mathfrak c}^ \#$$-polarized HBAV of relative dimension $$g$$, respectively all $${\mathfrak c}^ \#$$-polarized HBAV of relative dimension $$g$$ with principal level-$$n$$-structure. $${\mathcal M}$$, $${\mathcal M}({\mathfrak c}^ \#)$$, $${\mathcal M}_ n({\mathfrak c}^ \#)$$ are all algebraic stacks of finite type over $$\text{Spec} \mathbb{Z}$$. If $$n\geq 3$$, $${\mathcal M}_ n({\mathfrak c}^ \#)$$ is in fact a scheme; it is smooth of relative dimension $$g$$ and faithfully flat over $$\text{Spec} \mathbb{Z}[1/n,\zeta_ n]$$. Let $$\{\sigma^ C_ \alpha\}_{C,\infty}$$ be a $$\Gamma(n)$$-admissible polyhedral cone decomposition, where $$C$$ runs through all cusps of $${\mathcal M}_ n({\mathfrak c}^ \#)$$ and $$\alpha$$ runs through a certain indexing set $$I_ C$$. Let $$\tilde{\mathcal M}_ n({\mathfrak c}^ \#):={\mathcal M}_ n({\mathfrak c}^ \#)(\{\sigma^ C_ \alpha\})$$. – The main result of arithmetic toroidal compactification, due to Rapoport, can be summarized as follows: there is an open immersion of algebraic stacks $$j:{\mathcal M}_ n({\mathfrak c}^ \#)\subset\tilde{\mathcal M}_ n({\mathfrak c}^ \#)$$, a semi- abelian scheme $$G$$ over $$\tilde{\mathcal M}_ n({\mathfrak c}^ \#)$$, and a homomorphism $$m:{\mathfrak O}_ F\to\text{End}(G/\tilde{\mathcal M}_ n({\mathfrak c}^ \#))$$ such that $$G$$ extends the universal family over $${\mathcal M}_ n({\mathfrak c}^ \#)$$. The author points out that Rapoport’s theorem depends on unpublished results on rigid geometry, due to Raynaud. Another way to construct the arithmetic toroidal compactification of $${\mathcal M}_ n({\mathfrak c}^ \#)$$ is given in the book by G. Faltings and the author, “Degeneration of abelian varieties” (1990; Zbl 0744.14031).
In the paper under review, the author develops the arithmetic theory of minimal compactification of $${\mathcal M}_ n({\mathfrak c}^ \#)$$ which is an over-$$\mathbb{Z}$$-version of the Satake-Baily-Borel compactification for arithmetic quotients of bounded symmetric domains. For that purpose, he defines a line bundle $\omega_{\tilde{\mathcal M}_ n({\mathfrak c}^ \#)}=\bigwedge^ g(\underline{\text{Lie}}(G/\tilde{\mathcal M}_ n({\mathfrak c}^ \#)^ \lor)$ and considers its restriction to $${\mathcal M}_ n({\mathfrak c}^ \#)$$. By definition, a Hilbert modular form of level $$n$$ and weight $$k$$ is a global section of the sheaf $$\omega^{\otimes k}$$ over $${\mathcal M}_ n({\mathfrak c}^ \#)$$. — Some of the main facts proved in the paper are the following:
(i) $$\Gamma(\tilde{\mathcal M}_ n({\mathfrak c}^ \#),\omega^{\otimes k})=\Gamma({\mathcal M}_ n({\mathfrak c}^ \#),\omega^{\otimes k})$$, so that the first space does not depend on the choice of the $$\Gamma(n)$$- admissible polyhedral cone decomposition at the cusps.
(ii) A $$q$$-expansion principle for any $$\mathbb{Z}[\zeta_ n,1/n]$$-algebra, any $$k\in\mathbb{Z}$$, and any cusp $$C$$.
(iii) If $$n\geq 3$$, $$\omega_{\tilde{\mathcal M}_ n({\mathfrak c}^ \#)}$$ is generated by its global sections. Then, the canonical morphism $$\overline\pi:\tilde{\mathcal M}_ n({\mathfrak c}^ \#)\to\overline{\mathcal M}_ n({\mathfrak c}^ \#):=\text{Proj}_{\mathbb{Z}[\zeta_ n,1/n]}(\oplus_{k\geq 0}\Gamma(\tilde{\mathcal M}_ n({\mathfrak c}^ \#),\omega^ k))$$ is surjective and $$\overline M_ n({\mathfrak c}^ \#)$$ is a projective normal scheme of finite type over $$\mathbb{Z}[\zeta_ n,1/n]$$.
(iv) $$\overline M_ n({\mathfrak c}^ \#)$$ has a dense subscheme $$M_ n({\mathfrak c}^ \#)$$ which can be identified as the coarse moduli space of $${\mathcal M}_ n({\mathfrak c}^ \#)$$. The connected components of $$\overline M_ n({\mathfrak c}^ \#)\backslash M_ n({\mathfrak c}^ \#)$$ are in one-to- one correspondence with the cusps of $${\mathcal M}_ n({\mathfrak c}^ \#)$$.
(v) When $$n\geq 3$$, $$\omega\mid_{M_ n({\mathfrak c}^ \#)}$$ extends to an ample invertible sheaf $$\omega\mid_{\overline M_ n({\mathfrak c}^ \#)}$$ on $$\overline M_ n({\mathfrak c}^ \#)$$.
Two applications of the arithmetic theory of minimal compactifications are given in the paper. The first one concerns the existence of Hilbert modular forms, integral over $$\mathbb{Z}[\zeta_ m]$$ and with nebentypus character $$\chi:({\mathfrak O}/f{\mathfrak O})^ \times\to\mu_ m$$, the constant terms of which at unramified cusps take any pre-assigned values in $$\mathbb{Z}[\zeta_ m]$$. — The second application yields a geometric proof of a result due to Ribet which says that when $$\kappa$$ is an algebraically closed field of characteristic $$p>0$$, the $$p$$-adic monodromy attached to the étale quotient of the Barsotti-Tate group $$G[p^ \infty]/\kappa$$ over any component of the ordinary locus of $${\mathcal M}_ n({\mathfrak c}^ \#)/\kappa$$ is equal to $$({\mathfrak O}_ F\otimes_ \mathbb{Z}\mathbb{Z}_ p)^*$$, i.e., it is as large as possible.

##### MSC:
 14K10 Algebraic moduli of abelian varieties, classification 11R23 Iwasawa theory 14G35 Modular and Shimura varieties
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