Marius van der Put

Grothendieck's conjecture for the Risch equation y′ = ay + b

Abstract A simple formulation of the Grothendiecks conjecture, some information on p-curvatures, recent history and elementary proofs for the equations y′ = ay and y′ = b are given in the first two sections. For an inhomogeneous equation y′ = ay + b we propose an extension of the problem. On has to distinguish three cases. A proof, using Elkies result on supersingular primes for elliptic curves, covers part of the first case. The second case has a negative answer. The final case is shown to be related with recent progress in sieve theory. Some examples of degree two can be handled in this way.

Mumford coverings of the projective line

Abstract. A Mumford covering of the projective line over a complete non-archimedean valued field K is a Galois covering

Covers of Algebraic Curves

Xrightarrow {bf P}^1_K

Curves and Their Reductions

such that X is a Mumford curve over K. The question which finite groups do occur as Galois group is answered in this paper. This result is extended to the case where

The Projective Line

{bf P}^1_K

Etale Cohomology of Rigid Spaces

is replaced by any Mumford curve over K.

Valued Fields and Normed Spaces

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: n n(i) n np(G) is the smallest normal subgroup such that the factor group G/p(G) has no elements with order p. n n n n nii n np(G) is generated by all the p-Sylow subgroups of G.

Iterative differential equations and the Abhyankar conjecture

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} /leftlangle q rightrangle ) (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.

DIFFERENTIAL GALOIS THEORY, UNIVERSAL RINGS AND UNIVERSAL GROUPS

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.

Arithmetic and Differential Galois Groups

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.