Counting plane Mumford curves

TL;DR

This paper explores the enumeration of Mumford curves within a $p$-adic framework, linking tropical geometry, string theory, and $p$-adic Gromov-Witten invariants to identify configurations where all such curves are Mumford curves.

Contribution

It introduces methods to determine point configurations in $ ext{P}^2$ where all genus $g$, degree $d$ $p$-adic curves are Mumford curves, using tropical geometry techniques.

Findings

Identifies specific point configurations where all curves are Mumford curves.

Connects $p$-adic geometry with string theory and tropical geometry.

Provides a framework for counting Mumford curves in a $p$-adic setting.

Abstract

A $p$-adic version of Gromov-Witten invariants for counting plane curves of genus $g$ and degree $d$ through a given number of points is discussed. The multiloop version of $p$-adic string theory considered by Chekhov and others motivates us to ask how many of these curves are Mumford curves, i.e.\ uniformisable by a domain at the boundary of the Bruhat-Tits tree for $\PGL_2(\mathbb{Q}_p)$. Generally, the number of Mumford curves depends on the position of the given points in $\mathbb{P}^2$. With the help of tropical geometry we find configurations of points through which all curves of given degree and genus are Mumford curves. The article is preceded by an introduction to some concepts of $p$-adic geometry and their relation to string theory.