A refined Chabauty--Coleman bound for surfaces

TL;DR

The paper extends explicit Chabauty–Coleman methods to bound and sometimes determine the set of unexpected quadratic points on genus $3$ hyperelliptic curves by analyzing the surface $W_2 = C+C$ inside the Jacobian when the Mordell–Weil rank is at most one. It refines the Caro–Pastens approach to obtain a practical, prime-dependent bound at a suitable $p$, and integrates Siksek’s residue-disk criterion to certify equality with a computable finite set. The authors provide an explicit algorithm that combines the local $p$-adic analysis with global reduction data to bound $|W_2(\mathbb{Q})|$ and, in several examples, to determine $W_2(\mathbb{Q})$ exactly. These results improve the feasibility of determining $W_2(\mathbb{Q})$ for certain genus $3$ curves and illustrate how the Caro–Pastens framework can outperform existing methods in cases where Siksek’s criterion alone is not directly applicable.

Abstract

Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of rational points on the surface $W_2 := C+C$ in the case of a hyperelliptic curve $C$ of genus $3$ over $\mathbb{Q}$. Combining this with work of Siksek, we use this to determine $W_2(\mathbb{Q})$ in a selection of examples.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1: Method of Caro--Pasten
• Remark 2.2
• Definition 2.3: $\omega$-integrality
• Definition 2.4
• Proposition 2.5
• proof
• Theorem 2.6
• Remark 2.8