zbMATH — the first resource for mathematics

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 The author points out that Rapoport’s theorem depends on unpublished results on rigid geometry, due to Raynaud.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 √©tale 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.

14K10 Algebraic moduli of abelian varieties, classification
11R23 Iwasawa theory
14G35 Modular and Shimura varieties
Full Text: DOI