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 C₃ 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.