Algebraic curves with a large cyclic automorphism group

The study of algebraic curves \(\cX\) with numerous automorphisms in relation to their genus \(g(\cX)\) is a well-established area in Algebraic Geometry. In 1995, Irokawa and Sasaki \cite{Sasaki} gave a complete classification of curves over \(\mathbb{C}\) with an automorphism of order \(N \geq 2g(\mathcal{X}) + 1\). Precisely, such curves are either hyperelliptic with \(N=2g(\cX)+2\) with \(g(\cX)\) even, or are quotients of the Fermat curve of degree \(N\) by a cyclic group of order \(N\). Such a classification does not hold in positive characteristic \(p\), the curve with equation \(y^2=x^p-x\) being a well-studied counterexample. This paper successfully classifies curves with a cyclic automorphism group of order \(N\) at least \(2g(\mathcal{X}) + 1\) in positive characteristic \(p \neq 2\), offering the positive characteristic counterpart to the Irokawa-Sasaki result. The possibility of wild ramification in positive characteristic has presented a few challenges to the investigation.