Fumiharu Kato

Log smooth deformation and moduli of log smooth curves

The aim of this paper is to define a reasonable moduli theory of log smooth curves which recovers the classical Deligne–Knudsen–Mumford moduli of pointed stable curves.

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

A Mumford curve of genus g (>1) over a non-archimedean valued field k of positive characteristic has at most max{12(g-1), 2 g^(1/2) (g^(1/2)+1)^2} automorphisms. This bound is sharp in the sense that there exist Mumford curves of arbitrary high genus that attain it (they are fibre products of suitable Artin-Schreier curves). The proof provides (via its action on the Bruhat-Tits tree) a classification of discontinuous subgroups of PGL(2,k) that are normalizers of Schottky groups of Mumford curves with more than 12(g-1) automorphisms. As an application, it is shown that all automorphisms of the moduli space of rank-2 Drinfeld modules with principal level structure preserve the cusps.

Foundations of Rigid Geometry I

In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian (cf. introduction). The manuscript is encyclopedic and almost self-contained, and contains plenty of new results. A discussion on relationship with J. Tates rigid analytic geometry, V. Berkovichs analytic geometry and R. Hubers adic spaces is also included. As a model example of applications, a proof of Nagatas compactification theorem for schemes is given in the appendix.

Mumford curves with maximal automorphism group

1. IntroductionIt is well-known (cf. Conder [2]) that if a compact Riemann surface of genusg ≥ 2 attains Hurwitz’s bound 84(g − 1) on its number of automorphisms, thenits automorphism group is a so-called Hurwitz group, i.e., a finite quotient of thetriangle group ∆(2,3,7). Equivalently, the Riemann surface is an ´etale cover of theKlein quartic X(7). However, it is hitherto unknown which finite groups can occuras Hurwitz groups. It is even true that, for every integer n, there exists a g such thatthere are more than n non-isomorphic Riemann surfaces of genus g which attainHurwitz’s bound (Cohen, [1]). In this note, we want to show that the correspondingquestions for Mumford curves of genus g over non-archimedean valued fields ofpositive characteristic have a very easy answer: the maximal automorphism groupscan be explicitly described, and they occur for an explicitly given 1-parameterfamily of curves (at least for g /∈ {5,6,7,8}).The set-up for our result is as follows. Let (k,|·|) be a non-archimedean valuedfield of positive characteristic, and X a Mumford curve ([7], [5]) of genus g over k.This means that the stable reduction of X over the residue field ¯k of k is a unionof rational curves intersecting in ¯k-rational points. Equivalently, as a rigid analyticspace over k, the analytification X

Zur Entartung schwach verzweigter Gruppenoperationen auf Kurven

Abstract An action of a finite group on a smooth projective curve over an algebraically closed field of positive characteristic is called weakly ramified, if all second ramification groups are trivial (e.g., every group action on an ordinary curve is weakly ramified). When the ramification indices satisfy certain numerical criteria, we construct a degenerating equivariant quasi-projective family to which the given curve belongs, and which in a sense is the unique building block for all such weakly ramified equivariant families that ramify above a fixed set of points. The result is used to inductively study automorphisms of ordinary curves.