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Cyclic Cubic Points on Higher Genus Curves

James Rawson

TL;DR

This work investigates cyclic cubic points on curves over number fields, refining the Abramovich–Harris framework by incorporating Galois-group data. It proves that for genus $g \ge 5$, infinitely many cyclic cubic points occur exactly when there exists a degree-3 morphism $f:X\to Y$ with $Y$ being $\mathbb{P}^1$ or a positive-rank elliptic curve and with the discriminant curve $Y_{\Delta(f)}$ of the same type, providing computable tests to detect such morphisms. The paper also extends the analysis to lower genus under geometric or arithmetic hypotheses, and studies integral points via Levin-type results, yielding concrete finiteness criteria and applying them to modular curves such as $X_0(22)$ and $X_0(43)$. The results are supported by a careful use of Castelnuovo–Severi, resolution of gaps in previous proofs, and explicit examples illustrating the necessity of the discriminant-curve condition. Overall, the work clarifies when infinite families of cyclic cubic points arise and offers practical criteria for detecting the underlying degree-3 maps.

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.

Cyclic Cubic Points on Higher Genus Curves

TL;DR

This work investigates cyclic cubic points on curves over number fields, refining the Abramovich–Harris framework by incorporating Galois-group data. It proves that for genus , infinitely many cyclic cubic points occur exactly when there exists a degree-3 morphism with being or a positive-rank elliptic curve and with the discriminant curve of the same type, providing computable tests to detect such morphisms. The paper also extends the analysis to lower genus under geometric or arithmetic hypotheses, and studies integral points via Levin-type results, yielding concrete finiteness criteria and applying them to modular curves such as and . The results are supported by a careful use of Castelnuovo–Severi, resolution of gaps in previous proofs, and explicit examples illustrating the necessity of the discriminant-curve condition. Overall, the work clarifies when infinite families of cyclic cubic points arise and offers practical criteria for detecting the underlying degree-3 maps.

Abstract

The distribution of degree 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: . For curves of genus at least 5, we show cubic points with Galois group 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.
Paper Structure (8 sections, 32 theorems, 3 equations)

This paper contains 8 sections, 32 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Conventions
  4. Maps to $\mathbb{P}^1$
  5. A Theorem of Abramovich and Harris
  6. Genus $\geq 5$ curves
  7. Curves with genus 2, 3 or 4
  8. Integral points

Key Result

Theorem 1.1

Let $X / K$ be a curve of genus $g \geq 5$, then $X$ has infinitely many cyclic cubic points if and only if there exists a degree 3 morphism $f : X \to Y$ such that $Y$ is a $\mathbb{P}^1$ or a positive rank elliptic curve, and the "discriminant curve" of $f$ (see discrim) is also a $\mathbb{P}^1$ o

Theorems & Definitions (58)

  • Example 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.1
  • proof
  • Definition 2.2
  • ...and 48 more