Cyclic Cubic Points on Higher Genus Curves

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.

Comments welcome (14 pages). Version appearing in the Journal of the London Mathematical Society. Added a section to address deficiencies in Abramovich--Harris 1991, and re-structured the section on integral points to incorporate Levin 2016