Constraints on the automorphism group of a curve

TL;DR

This paper introduces a criterion to determine the non-existence of certain automorphisms in high-genus curves over finite fields and applies it to identify automorphism groups of specific modular curves.

Contribution

It provides a new criterion for ruling out automorphisms of prime power order and demonstrates its application to modular curves of high genus.

Findings

Criterion effectively excludes automorphisms of prime power order.

Applied criterion to determine automorphism groups of certain modular curves.

Enhanced understanding of automorphism structures in high-genus curves.

Abstract

For a curve $X$ of genus $>1$ defined over a finite field, we present a criterion which allows us to state the non existence of automorphisms of order a power of a rational prime. We show how this criterion can be used to determine the automorphism group of some modular curves of high genus.