# The relation between rigid-analytic and algebraic deformation parameters for Artin-Schreier-Mumford curves

@article{Cornelissen2010TheRB, title={The relation between rigid-analytic and algebraic deformation parameters for Artin-Schreier-Mumford curves}, author={Gunther Cornelissen and Fumiharu Kato and Aristides Kontogeorgis}, journal={Israel Journal of Mathematics}, year={2010}, volume={180}, pages={345-370} }

We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk.

## 5 Citations

Measure-theoretic rigidity for Mumford curves

- MathematicsErgodic Theory and Dynamical Systems
- 2012

Abstract One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a…

A combinatorial Li–Yau inequality and rational points on curves

- Mathematics
- 2012

We present a method to control gonality of nonarchimedean curves based on graph theory. Let $$k$$k denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a…

Berkovich curves and Schottky uniformization.

- Mathematics
- 2020

This text is an exposition of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. It consists of two parts, the first one of an introductory nature, and…

Aspects of p-adic geometry related to entanglement entropy

- Computer Science
- 2020

The purpose of this survey paper is to outline some aspects of p-adic geometry that are naturally related to the problem of formulating an analog of the usual replica argument for entanglement entropy in the context ofp-adic holography.

## References

SHOWING 1-10 OF 28 REFERENCES

Schottky Groups and Mumford Curves

- Mathematics
- 1980

Discontinuous groups.- Mumford curves via automorphic forms.- The geometry of mumford curves.- Totally split curves and universal coverings.- Analytic reductions of algebraic curves.- Jacobian…

Rigid analytic geometry and its applications

- Mathematics
- 2003

Preface.- Valued fields and normed spaces.- The projective line.- Affinoid algebras.- Rigid spaces.- Curves and their reductions.- Abelian varieties.- Points of rigid spaces, rigid cohomology.- Etale…

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…

A Hauptsatz of L. E. Dickson and Artin-Schreier extensions.

- Mathematics
- 1980

Let K be an algebraically closed field. If F/K is a hyperelliptic function field, it contains a unique rational subfield K(x) of index 2. (K(x) is generated by the quotients of holomorphic…

Cyclic coverings of the p-adic projective line by Mumford curves

- Mathematics
- 2007

Exact bounds for the positions of the branch points for cyclic coverings of the p-adic projective line by Mumford curves are calculated in two ways. Firstly, by using Fumiharu Kato’s *-trees, and…

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,…

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

- Mathematics
- 1999

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…

$p$-ranks and automorphism groups of algebraic curves

- Mathematics
- 1987

Let X be an irreducible complete nonsingular curve of genus g over an algebraically closed field k of positive characteristic p. If 9 > 2, the automorphism group Aut(X) of X is known to be a finite…

The p-rank of Artin-Schreier curves

- Mathematics
- 1975

The groundfield k is algebraically closed and of characteristic p ≠ O. The p-rank of an abelian variety A/k is σA if there are σA copies of Z/pZ in the group of points of order p in A(k). The p-rank…

Periods of p-adic Schottky groups.

- Mathematics
- 1973

An element g £ G is called hyperbolic, if it is represented by a matrix whose characteristic roots have different absolute values. A subgroup Γ < G is called Schottky group, if it is finitely…