Jean Fresnel

Covers of Algebraic Curves

Let p be a prime number and let G be a finite group. The subgroup p(G) ⊂ G is defined as the group generated by all elements in G having as order a power of p. Equivalent definitions of p(G) are: (i) p(G) is the smallest normal subgroup such that the factor group G/p(G) has no elements with order p. ii p(G) is generated by all the p-Sylow subgroups of G.

Curves and Their Reductions

Let k be a complete non-archimedean valued field and q ∈ k with \( 0 for the subgroup of k* generated by q. The elements in are seen as automorphisms of \( G_{m,k}^{an} \). The Tate curve is the object \( \mathcal{T}: = G_{m,k}^{an} /\left\langle q \right\rangle \) (we keep this somewhat heavy notation in order to avoid confusions which might arise from the notation k* / ). In the sequel we will explain the rigid analytic structure of \( \mathcal{T} \), compute the field of meromorphic functions on it and show that \( \mathcal{T} \) is the analytification of an elliptic curve over k of a special type.

The Projective Line

In this chapter K is an algebraically closed field, complete with respect to a nonarchimedean valuation. The projective line P over K is, as usual, K ⋃ {∞} and z denotes the variable of P. The goal of this chapter is to develop the function theory on suitable subsets of P with a minimum of mathematical tools. This elementary level is already sufficient for the exposition of D. Harbater’s theorem, [125], concerning the Galois groups of extensions of the field Q p (z). Moreover, the rigid analytic part of M. Raynaud’s proof of Abhyankar’s conjecture for the affine line in positive characteristic, [213], can be understood without further knowledge of rigid analytic spaces. We will present these proofs in Chapter 9.

Etale Cohomology of Rigid Spaces

Fundamental groups, etale topology and etale cohomology have been conceived in algebraic geometry as a theory which captures topological properties, well known for real and complex varieties. Especially for algebraic varieties over a field of positive characteristic, this theory produces surprising analogies with the algebraic topology of real or complex varieties. One of the early successes is of course the proof of the Weil conjectures. For rigid spaces, a first glimpse of etale cohomology appeared in the work of Drinfel’d on the Langlands conjectures for function fields in positive characteristic. For certain rigid spaces, e.g., the Drinfel’d symmetric spaces Ω(n) (see Examples 7.4.9) and more general symmetric spaces, etale cohomology forms the basis for the study of automorphic representations and Galois representations. There are many names attached to these “p-adic theories” (P. Schneider, U. Stuhler, M. Rapoport, P. Deligne, Th. Zink et al.) Etale cohomology for rigid spaces is developed by V.G. Berkovich, O. Gabber (unpublished), R. Huber, A.J. de Jong, M. van der Put, K. Fujiwara et al. Berkovich, in the paper [21], develops an etale cohomology for “analytic spaces”. This category of analytic spaces was introduced in [22] and extended in [21]. It is different from the category of rigid spaces. For this reason we will not borrow from his work. However, we have to mention that the approach taken here, in some sense, does not differ from his. Furthermore, using the equality of Berkovich cohomology with the one presented here in the case of paracompact varieties (see [143], Section 8.3), all the results presented here are in principle deducible from the references [21, 20, 19, 18]. R. Huber constructed an etale cohomology theory for his adic spaces. This theory specializes to a theory for rigid spaces, too.

Valued Fields and Normed Spaces

A valued field is a field k provided with a valuation (or absolute value). The latter is a map a ↦ |a| from k to R which satisfies the rules:

Rigid analytic geometry and its applications

\begin{gathered} 1. \left| a \right| \geqslant 0 and \left| a \right| = 0 if and only if a = 0. \hfill \\ 2. \left| {ab} \right| = \left| a \right| \cdot \left| b \right|. \hfill \\ 3. \left| {a + b} \right| \leqslant \left| a \right| + \left| b \right|. \hfill \\ \end{gathered}

Algèbres

The absolute values on the fields R and C make these fields into valued fields. In the present context the most important valued fields are the so-called nonarchimedean valued fields. They are defined by replacing the triangle inequality \( \left| {a + b} \right| \leqslant \left| a \right| + \left| b \right| \) with the stronger inequality \( \left| {a + b} \right| \leqslant \max \left( {\left| a \right|,\left| b \right|} \right) \). The trivial valuation is defined by \( \left| 0 \right| = 0 and \left| a \right| = 1 for a \in k^* \). In what follows we will mean by valuation and valued field, a non-trivial, non-archimedean valuation and a field equipped with such a valuation.

L^1

A basic observation concerning abelian sheaves on a rigid space X is that the set of its ordinary points is too small (e.g., Exercise 7.0.11). In particular, there are abelian sheaves \( \mathcal{F} \ne 0 \) on X such that the stalk \( \mathcal{F}_x \) is 0 for every x ∈ X. The obvious reason is that the Grothendieck topology on X is not local enough. The first concept of a sufficient collection of points for a rigid space is presented in [198]. This concept, its generalizations and rigid etale cohomology has been developed by V.G. Berkovich, O. Gabber (unpublished), P. Schneider, R. Huber, A.J. de Jong, K. Fujiwara et al. Especially, V.G. Berkovich has build an extensive theory of “non-archimedean analytic spaces” on this concept and a complete theory of even more general “adic spaces” is the work of R. Huber. Here we will give an introduction for this concept of points and prove some of the basic results. We will however, not leave the framework of rigid spaces.