$p$-adic Hurwitz numbers

TL;DR

This paper introduces stable tropical curves to count $p$-adic covers of the projective line by Mumford curves using tropical Hurwitz numbers, linking algebraic and tropical geometry.

Contribution

It develops a new method to compute $p$-adic Hurwitz numbers via tropical geometry, specifically through stable tropical curves and branch locus dependence.

Findings

Counts of $p$-adic covers expressed in tropical Hurwitz numbers

Dependence of counts on branch loci

Establishment of a tropical approach to $p$-adic enumerations

Abstract

We introduce stable tropical curves and use these to count covers of the $p$-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers.