The explicit Mordell Conjecture for families of curves (with an appendix by M. Stoll)

TL;DR

This paper proves the explicit Mordell Conjecture for large families of curves and introduces a practical method for computing all rational points on various curves, supported by explicit examples and comparisons.

Contribution

It provides an explicit proof of the Mordell Conjecture for broad classes of curves and develops an easy-to-apply computational method for rational points.

Findings

Successful computation of rational points on diverse curves

Explicit bounds enable effective computer searches

Comparison with classical methods highlights advantages

Abstract

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin-Demjanenko and the analysis of our explicit examples is carried to conclusion.