Abelian Varieties from the Rigid Analytic Viewpoint

Abstract

This chapter describes the Abelian varieties from the rigid analytic viewpoint. A substantial part of Barsottis work on Abelian varieties was motivated by the problem of determining their structure. The geometric description of Abelian varieties in terms of uniformizing spaces and lattices plays a central role in compactifying moduli spaces of Abelian varieties. To compactify, the idea is to enlarge to a stack in such a way that the valuative criterion of properness holds. It is found that if one succeeds to add the necessary semi-Abelian schemes at the boundary, one can expect that the valuative criterion of properness will work, thus, obtaining a compactification. Adding semi-Abelian schemes at the boundary is not an easy task. Polarizations as occurring have to be parameterized in a very careful way. All this has to be done over complete noetherian normal rings, which are more general than just valuation rings. It is found that doing so, either proceeding through formal schemes or adapting rigid geometry to a complete noetherian base, much of the geometric significance of degenerations present over a valuation ring is lost.