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Explicit reduction theory for Siegel modular threefolds. (English) Zbl 0789.11029

In this long and very explicit paper the authors construct a retract \(W\) of the Siegel upper half plane \(\mathbb{H}_ 2\) of degree two and a decomposition of \(W\) into cells \(B(X) \subset \mathbb{H}_ 2\) indexed by a partially ordered set \((W,\ll)\) such that \(W\) becomes a regular, locally finite, contractible cell-complex. Moreover, if \(\Gamma\) is a torsion- free subgroup of the Siegel modular group \(Sp(4,\mathbb{Z})\) of finite index, the complex \(W\) is a \(\Gamma\)-equivalent deformation retract of \(\mathbb{H}_ 2\) with finitely many \(\Gamma\)-orbits, hence \(\mathbb{H}_ 2/ \Gamma\) has the same homotopy type as the finite cell-complex \(W/ \Gamma\) which is in fact homeomorphic to the order complex of \((W/ \Gamma,\ll)\). Thus the topology of \(W/ \Gamma\) is reflected by the poset \((W,\ll)\) which is given quite explicitly by elements of certain subsets of a symplectic lattice – intimately connected with Voronoi’s reduction theory of \(GL(4,\mathbb{R})\) – and reverse inclusion as order relation.
To be more precise, first a decomposition of \(\mathbb{H}_ 2\) into cells \(M(X)\), \(X \in W\), is achieved so to form an analytic Whitney- stratification of \(\mathbb{H}_ 2\) compatible with the action of \(\Gamma\) and with finitely many orbits. This decomposition is nothing but the Poincaré dual decomposition of \(W\) into cells \(B(X)\), i.e. the cells \(M(X)\) and \(B(X)\) meet each other transversely at one point and their dimensions add up. In order to decompose \(\mathbb{H}_ 2\) into cells, this space is embedded into the larger linear symplectic space \(SO(4,\mathbb{R})/SL(4,\mathbb{R})\) for which the Voronoi decomposition is the analogue of the classical decomposition of the upper half plane \(\mathbb{H}\) into hyperbolic triangles. The intersection of these Voronoi cells with \(\mathbb{H}_ 2\) furnishes the required decomposition. Since there is no linear structure on \(\mathbb{H}_ 2\) in contrast to the classical elliptic case \(\mathbb{H}\) the construction of the cells \(B(X)\) causes serious problems which are surpassed by extending the decomposition of \(\mathbb{H}_ 2\) to a decomposition of a Satake partial compactification \(\mathbb{H}^*_ 2\) of \(\mathbb{H}_ 2\) having the same geometric properties.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
14J30 \(3\)-folds
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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References:

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