Artin-Schreier Root Stacks

TL;DR

This paper introduces Artin-Schreier root stacks to classify stacky curves in characteristic p, enabling direct incorporation of ramification data and facilitating computations of Riemann-Roch spaces and modular forms.

Contribution

It develops a new framework using Artin-Schreier root stacks to classify and analyze stacky curves with cyclic stabilizers in characteristic p, replacing traditional local structures.

Findings

Classified stacky curves with cyclic stabilizers using ramification data.

Computed dimensions of Riemann-Roch spaces for specific stacky curves.

Proposed a program for computing modular forms in positive characteristic.

Abstract

We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin-Schreier root stack, allowing us to incorporate this ramification data directly into the stack. As an application, we compute dimensions of Riemann-Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms in this setting.