Werner Lütkebohmert

Formal and rigid geometry. I. Rigid spaces.

In 1974 Raynaud proposed a program (Raynaud in Mem. Soc. Math. Fr. 39–40:319–327, 1974), where he introduced groundbreaking ideas to rigid geometry by interpreting a rigid analytic space as the generic fiber of a formal schemes over \(\operatorname{Spf}R\). Here \(\operatorname{Spf}R\) is always the formal spectrum of a complete valuation ring \(R\) of height 1, where its topology is given by an ideal \((\pi)\) for some element \(\pi\in R\) with \(0<|\pi|<1\). Due to results on flat modules (Raynaud and Gruson in Invent. Math. 13:1–89, 1971) his approach also works in the non-Noetherian case of formal schemes of topologically finite presentation over \(\operatorname{Spf}(R)\).

Rigid geometry of curves and their Jacobians

This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.

What Is a Néron Model

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

Properties of Néron Models

Although the notion of a Neron model is functorial, it cannot be said that Neron models satisfy the properties, one would expect from a good functor. For example, Neron 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 Neron models with semi-abelian reduction.

Rigid Analytic Curves

The main objective of this chapter is the Stable Reduction Theorem 4.5.3 for smooth projective \(K\)-curves \(X_{K}\). Its proof is split into two problems. In a first step, dealt with in Sect. 3.4, we provide a projective \(R\)-model \(X\) of \(X_{K}\) such that its special fiber \(X\otimes_{R}k\) is reduced. In a second step we will now analyze the singularities of \(X\otimes _{R}k \). This part is related to the resolution of singularities in dimension 2.

Classical Rigid Geometry

In this chapter we give a survey of rigid geometry over non-Archimedean fields. The foundation of the theory was laid by Tate in his private Harvard notes dating back to 1961, which were later published in Inventiones mathematicae (Tate in Invent. Math. 12:257–289, 1971). Here we explain the main results from the classical point of view as studied in the late sixties; for proofs we refer to Bosch (Lectures on Formal and Rigid Geometry, vol. 2105, 2014). At that time rigid geometry was mainly inspired by complex analysis. Fundamental results were achieved by Kiehl, who introduced the Grothendieck topology and proved the basic facts concerning coherent sheaves. Moreover, Kiehl makes Serre’s theory (Serre in Ann. Inst. Fourier 6:1–42, 1956) of Geometrie Algebrique et Geometrie Analytique available for rigid analytic geometry, often referred to as GAGA; cf. (Kopf in Uber eigentliche Familien algebraischer Varietaten uber affinoiden Raumen, vol. 7, 1974).

&#201TALE COVERS OF A PUNCTURED p -ADIC DISC

The aim of this paper is to report on recent work on éetale covers of the punctured disc. The paper surveys basic results on curves over p-adic fields and explains ideas of Riemanns existence problem for a p-adic field giving full details and as well new results. In [17] the problem of the extension of an étale cover φ* : X* → D* of a punctured disc D*, defined over an p-adic field K, to a (ramified) cover φ : X → D was proved in the case where the base field K has characteristic 0. The behavior of the discriminant of such an étale cover is now well understood. Moreover, new results of the second author [22] in the case of positive characteristic char(K) > 0 are presented as well. The paper ends with two interesting examples of covers in positive characteristic which are not extendable.

The Smoothening Process

The smoothening process, in the form needed in the construction of Neron 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 Neron models under appropriate conditions.

Jacobians of Relative Curves

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 Neron model of the Jacobian J K of the generic fibre X K .

Some Background Material from Algebraic Geometry

In this chapter we give a review of some basic tools which are needed in later chapters for the construction of Neron models. Assuming that the reader is familiar with Grothendieck’s definition of schemes and morphisms, we treat the concept of smooth and etale 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].