Effective Mordell for curves with enough automorphisms

TL;DR

This work establishes fully explicit height bounds for rational points on curves of genus at least $2$ over number fields under a hypothesis of sufficiently large automorphism groups relative to the Jacobian's Mordell–Weil rank. The authors develop an Arakelov-theoretic framework, combining admissible metrics, Green functions, fibral corrections, and a canonical height on the Jacobian to derive an explicit bound $\hat{h}(j(P)) \le \frac{M(X)}{2g\tau}$ for points with trivial stabilizers, where $M(X)$ aggregates archimedean and non-archimedean contributions. A key innovation is the explicit Mumford gap principle, together with an effective procedure to compute fibral corrections and the associated $\phi_\mathfrak{p}(X)$ terms; these yield practical height bounds and enable complete determinations of $X(K)$ in explicit genus $2$ examples, as demonstrated on a curve with rank $2$ and $8$ automorphisms. The results provide a constructive alternative to Chabauty-type methods and illustrate how Arakelov theory can directly inform the determination of rational points on higher-genus curves in cases with rich automorphism structure.

Abstract

We prove a completely explicit and effective upper bound for the Néron--Tate height of rational points of curves of genus at least $2$ over number fields, provided that they have enough automorphisms with respect to the Mordell--Weil rank of their jacobian. Our arguments build on Arakelov theory for arithmetic surfaces. Our bounds are practical, and we illustrate this by explicitly computing the rational points of a certain genus $2$ curve whose jacobian has Mordell--Weil rank $2$.

Theorems & Definitions (32)

• Theorem 1.1: Main result
• Theorem 1.2: Explicit gap principle
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• proof
• Theorem 3.4