Cyclic cubic points on higher genus curves

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.