Purely additive reduction of abelian varieties with torsion

Abstract

Let OK be a discrete valuation ring with fraction field K of characteristic 0 and algebraically closed residue field k of characteristic p>0. Let A/K be an abelian variety of dimension g with a K-rational point of order p. In this article, we are interested in the reduction properties that A/K can have. After discussing the general case, we specialize to g=1, and we study the possible Kodaira types that can occur.