The Siegel moduli scheme classifies principally polarised abelian varieties and its compactification is an important result in arithmetic algebraic geometry. The main result of this monograph is to prove the existence of the toroidal compactification over Z (1/2). This result should have further applications and is presented here with sufficient background material to make the book suitable for seminar courses in algebraic geometry, algebraic number theory or automorphic forms.

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The central theme of this work is a proof of the irreducibility of the moduli space of principally polarized Abelian varieties in characteristic p greater than 2. The presentation is self-contained and assumes only a knowledge of basic geometry.

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