Néron Models

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.

This chapter is meant to provide a first orientation to the basics of Néron models. Among other things, it contains an explanation of the context in which Néron models are considered, as well as a discussion of the main results on the construction and existence, including some examples.

In this chapter we give a review of some basic tools which are needed in later chapters for the construction of Néron models. Assuming that the reader is familiar with Grothendieck’s definition of schemes and morphisms, we treat the concept of smooth and étale morphisms, of henselian rings, and of S-rational maps; moreover, we have included some facts on differential calculus and on flatness. Concerning the smoothness, we give a self-contained exposition of this notion, relating it closely to the Jacobi criterion. For the other topics we simply state results, sometimes without giving proofs. Most of the material presented in this chapter is contained in Grothendieck’s treatments [EGA IV₄] and [SGA 1].

The smoothening process, in the form needed in the construction of Néron models, is presented in Sections 3.1 to 3.4. After we have explained the main assertion, we discuss the technique of blowing-up which is basic for obtaining smoothenings. The actual proof of the existence of smoothenings is carried out in Sections 3.3 and 3.4. As an application, we construct weak Néron models under appropriate conditions.

In the previous chapter, we discussed the smoothening process and, as an application, proved the existence of weak Néron models. The next step towards the construction of Néron models requires the use of group arguments.

For the construction of Néron models, we need the fact that an S-birational group law on a smooth S-scheme with non-empty fibres can be birationally enlarged to a smooth S-group scheme; see 4.3/6. The purpose of the present section is to prove this result in the case where S is strictly henselian. In Chapter 6, the result will be extended to a more general base.

During the years 1959 to 1962, Grothendieck gave a series of six lectures at the Séminaire Bourbaki, entitled “Technique de descente et théorèmes d’existence en géométrie algébrique”. In the first lecture [FGA], n°190, the general technique of faithfully flat descent is introduced. It is an invaluable tool in algebraic geometry. Quite often it happens that a certain construction can be carried out only after faithfully flat base change. Then one can try to use descent theory in order to go back to the original situation one started with. Before Grothendieck, descent was certainly known in the form of Galois descent.

Although the notion of a Néron model is functorial, it cannot be said that Néron models satisfy the properties, one would expect from a good functor. For example, Néron models do not, in general, commute with (ramified) based change; also, in the group scheme case, the behavior with respect to exact sequences can be very capricious. The situation stabilizes somewhat if one considers Néron models with semi-abelian reduction.

Following Grothendieck’s treatment [FGA], we introduce the relative Picard functor Pic_X/S and treat the notion of the rigidified relative Picard functor. The main purpose of this chapter is the presentation of various results on the representability of Pic_X/S. We explain Grothendieck’s theorem on the representability of Pic_X/S by a scheme and point out improvements due to Mumford [2] as well as those due to Altman and Kleiman [1]. In Section 8.3, we discuss the main steps of M. Artin’s approach [5] to the representability of Pic_X/S by an algebraic space; for details, the reader is referred to his paper. At the end of the chapter, there is a collection of some results on smoothness as well as on finiteness properties of Pic_X/S, as can be found in [SGA 6].

The chapter consists of two parts. In the first four sections we study the represent-ability and structure of Pic _X/S for a relative curve X over a base S. Then, in the last three sections, we work over a base S consisting of a discrete valuation ring R with field of fractions K and, applying these results, we investigate the relationship between Pic _X/S and the Néron model of the Jacobian J _K of the generic fibre X _K .

For this last chapter we introduce a new type of Néron models, so-called Néron 1ft-models. To define them, we modify the definition of Néron models by dropping the condition that they are of finite type. Then, due to the smoothness, Néron lft-models are locally of finite type. This is the reason why we use the abbreviation “lft”. For example, tori do admit Néron lft-models whereas, for non-zero split tori, Néron models (in the original sense) do not exist.