# Discontinuous groups in positive characteristic and automorphisms of Mumford curves

@article{Cornelissen1999DiscontinuousGI, title={Discontinuous groups in positive characteristic and automorphisms of Mumford curves}, author={Gunther Cornelissen and Fumiharu Kato and Aristeides Kontogeorgis}, journal={Mathematische Annalen}, year={1999}, volume={320}, pages={55-85} }

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…

## 44 Citations

### Mumford curves with maximal automorphism group

- Mathematics
- 2002

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…

### On the non‐existence of exceptional automorphisms on Shimura curves

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- 2008

We study the group of automorphisms of Shimura curves X0(D, N) attached to an Eichler order of square‐free level N in an indefinite rational quaternion algebra of discriminant D>1. We prove that,…

### Mumford Curves with Maximal Automorphism Group II: Lamé Type Groups in Genus 5-8

- Mathematics
- 2002

A Mumford curve of genus g=5,6,7 or 8 over a non-Archimedean field ofcharacteristic p (such that if p=0, the residue field characteristic exceeds 5) has at most 12(g−1) automorphisms. In this paper,…

### Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

- Mathematics
- 2001

We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the "algebraic" deformation functor of an…

### Automorphisms of curves and Weierstrass semigroups for Harbater–Katz–Gabber covers

- MathematicsTransactions of the American Mathematical Society
- 2019

We study
p
p
-group Galois covers
X
→
P
1
X \rightarrow \mathbb {P}^1
with only one fully ramified point in characteristic
p
>
0
p>0
. These covers…

### automorphism groups for p-cyclic covers of the affine line

- MathematicsCompositio Mathematica
- 2005

let k be an algebraically closed field of positive characteristic p > 0 and $c \to {\mathbb p}^1_k$ a p-cyclic cover of the projective line ramified in exactly one point. we are interested in the…

### Graph Theoretic Construction of Discrete Groups over p-adic Fields

- Mathematics
- 2000

This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that…

### On Abelian Automorphism Groups of Mumford Curves and Applications to Shimura Curves

- Mathematics
- 2006

We use rigid analytic uniformization by Schottky groups to give a bound for the order of abelian subgroups of the automorphism groups of Mumford curves in terms of the genus. Furthermore, we give a…

### The Geometry of the Artin-Schreier-Mumford Curves over an Algebraically Closed Field

- Mathematics
- 2016

For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation…

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We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the "algebraic" deformation functor of an…

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