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Compactification of Siegel moduli schemes. (English) Zbl 0578.14009

London Mathematical Society Lecture Note Series, 107. Cambridge etc.: Cambridge University Press. XVI, 326 p. £15.00; $ 29.95 (1985).
This monograph (based on the author’s Harvard dissertation) is concerned with the problem of compactification of the moduli space of principally polarised (\(g\)-dimensional) abelian varieties over \(\mathbb{Z}\). For elliptic curves \(E\), with \(j\) = the \(j\)-invariant of \(E\), the ”\(j\)-line” \(\text{Spec}\mathbb{Z}[j]\) gives the moduli; the natural completion of the affine \(j\)-line \(\mathbb{A}^1\) is just \(\mathbb{P}^1\) and we have a ’canonical’ solution for the compactification problem. When \(g>1\), the isomorphism classes of principally polarized abelian varieties correspond to points of the (coarse) moduli space \(A_ g\) of such abelian varieties; Mumford solved the classification problem, constructing a coarse moduli scheme \(A_ g\) over Spec \(\mathbb{Z}\), via his geometric invariant theory. Associated with the moduli scheme \(A_ g\), there exists a fundamental geometric problem underlying the need for the compactification: namely, for a given prime number \(p\), \[ \begin{cases} \vtop{=.85 noindent is the geometric fibre \(A_ g \underset{\text{Spec}\mathbb{Z}}\times \text{Spec}\mathbb{F}_ p\) irreducible or equivalently, \smallskip\noindent is the moduli space of principally polarized abelian varieties irreducible, in characteristic \(p\)?} \end{cases} \tag{\(*\)} \] An affirmative answer to (\(*\)) for \(p>2\) is a consequence of the author’s construction of toroidal completions of Siegel moduli schemes over \(\mathbb{Z}[\frac12]\) and extension (to positive characteristics) of a theorem of Tai on the ”projectivity of toroidal compactifications” (substituting local holomorphic functions in Tai’s proof with the algebraic machinery of theta functions). Let \(M(\mathbb{Z},k)\) denote the ring of Siegel modular forms of degree \(g\) and weight \(k\) with Fourier coefficients in \(\mathbb{Z}\), and \(R\) the graded ring \(\oplus_{k\geq 0} M(\mathbb{Z},k)\). For \(g=1\), it is well-known that \(R\) is finitely generated. The corresponding question for general \(g\) was raised by J. Iqusa who also provided in a nice paper [Am. J. Math. 101, 149-183 (1979; Zbl 0415.14026)] an explicit set of generators for \(g=2\). The author’s affirmative answer to (\(*\)) above (for \(p>2\)) implies that the graded ring of Siegel modular forms with Fourier coefficients in \(\mathbb{Z}[]\) is finitely generated over \(\mathbb{Z}[]\). — A footnote on page xi refers to the question (\(*\)) of irreducibility for the case \(p=2\) having since been settled by Faltings.
Chapter I reviews the major results on Siegel moduli schemes used in subsequent chapters. The next chapter contains a treatment of semi-abelian varieties and with a vital definition of ”polarization”, the canonical construction of the quotient of a semi-abelian scheme by a discrete subgroup. These results are applied in chapter III to construct polarized semi-abelian schemes providing local coordinates, at the boundary, of toroidal completions of Siegel moduli schemes. Chapter IV contains the main result (theorem 4.2) of the author extending Tai’s theorem to positive characteristics, and chapter V contains nice applications to Siegel modular forms.
There are three excellent appendices (dealing with theta functions); the results in the appendix on 2-adic theta functions with values in complete local fields \(k\) with a uniformization theorem for abelian varieties over k (with residue characteristic \(\neq2)\) are unpublished results due to Mumford.
Reviewer: S.Raghavan

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14K10 Algebraic moduli of abelian varieties, classification
14L24 Geometric invariant theory
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
11F06 Structure of modular groups and generalizations; arithmetic groups
14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 0415.14026