Cyclic Cubic Points on Higher Genus Curves

TL;DR

This paper investigates the distribution of degree 3 points on higher genus curves, focusing on Galois groups and providing criteria for the existence of certain morphisms, extending understanding beyond low-genus cases.

Contribution

It refines the understanding of cubic points on higher genus curves by linking Galois groups to morphisms and offering computable tests for their existence.

Findings

Cubic points with Galois group C_3 are linked to specific morphisms on curves of genus ≥ 5.

Provides computable criteria for the existence of these morphisms.

Extends results to lower genus curves under additional assumptions.

Abstract

The distribution of degree $d$ points on curves is well understood, especially for low degrees. We refine this study to include information on the Galois group in the simplest interesting case: $d = 3$. For curves of genus at least 5, we show cubic points with Galois group $C_3$ arise from well-structured morphisms, along with providing computable tests for the existence of such morphisms. We prove the same for curves of lower genus under some geometric or arithmetic assumptions.