Introduction

Let K be a number field, S the spectrum of its ring of integers, and A _K an abelian variety over K. Standard arguments show that A _K extends to an abelian scheme A’ over a non-empty open part S” of S. Thus A _K has good reduction at all points s of S’ in the sense that A _K extends to an abelian scheme or, what amounts to the same, to a smooth and proper scheme over the local ring at s. In general, one cannot expect that A _K also has good reduction at the finitely many points in S — S’. However, one can ask if, even at these points, there is a notion of “good” models which generalizes the notion of good reduction. It came as a surprise for arithmeticians and algebraic geometers when A. Néron, relaxing the condition of properness and concentrating on the group structure and the smoothness, discovered in the years 1961–1963 that such models exist in a canonical way; see Néron [2], see also his lecture at the Séminaire Bourbaki [1]. Gluing these models with the abelian scheme A’, one obtains a smooth S-group scheme A of finite type which may be viewed as a best possible integral group structure over S on A _K . It is called a Néron model of A _K and is characterized by the universal property that, for any smooth S-scheme Z and any K-morphism u _K : Z _K → A _K , there is a unique S-morphism и : Z → A extending u _K . In particular, rational points of A _K can be interpreted as integral points of A.