Component Groups and Non-Archimedean Uniformization

Abstract

In this chapter, we will study the behaviour of the torsion part of the Néron component group of a semi-abelian K-variety under ramified extension of the base field K. Our main goal is to prove the rationality of the component series (Theorem 5.5.4.2). We discussed the case of an abelian K-variety in Halle and Nicaise (Math Ann 348(3):749–778, 2010); in that case, the component group is finite. The main complication that arises in the semi-abelian case is the fact that it is difficult in general to identify the torsion part of the component group in a geometric way. This problem is related to the index of the semi-abelian K-variety, an invariant that we introduce in Sect. 5.2. For tori, the torsion part of the component group has a geometric interpretation in terms of the dual torus, and we can explicitly compute the index from the character group.The case of a semi-abelian variety is substantially more difficult; there we need to construct a suitable uniformization, which is no longer an algebraic group but a rigid analytic group. In order to deal with Néron component groups of rigid analytic groups, we will use the cohomological theory of Bosch and Xarles (Math Ann 306:459–486, 1996), that we recall and extend in Sect. 5.1. We correct an error in their paper, which was pointed out by Chai, and we show that all of the principal results in Bosch and Xarles (Math Ann 306:459–486, 1996) remain valid.