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The siegel upperhalf plane and the symplectic group.- Main problem and main results.- The case of g=1.- Boundary components and the structure of parabolic subgroups.- Realization as a siegel domain of the third kind, and satake compactification.- Theory of torus embeddings.- Toroidal compactification due to Mumford.- Examples : reduction theory of positive quadratic forms.- An application of the Voronoi compactification to the theory of moduli of polarized abelian varieties.
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