A trick to ensure positive Mordell-Weil rank

TL;DR

This paper introduces a practical criterion to guarantee positive Mordell-Weil rank for Jacobians of smooth curves over number fields, with theoretical and computational implications, supported by explicit examples.

Contribution

It provides a new criterion linking rational divisor classes and Mordell-Weil rank, including refinements and explicit examples for practical use.

Findings

A smooth curve with no rational 2-torsion and theta characteristic has positive rank if it has a rational degree 1 class.

The criterion applies to generic curves and can be used in computational settings.

Explicit examples demonstrate the criterion's effectiveness.

Abstract

In this short note, we present a trick to ensure that the Jacobian of a given smooth curve over a number field has strictly positive Mordell-Weil rank. More explicitly, we prove that a smooth curve with no rational non-trivial 2-torsion and no rational theta characteristic has non-zero Mordell-Weil rank assuming the existence of a rational degree 1 divisor class. In particular, it implies that a generic nice curve with a rational degree 1 class has strictly positive rank. This criteria is both of theoretical and computational interest as we show how to use it in practice. We also give refinements, including an equivalent for families of curves, and explicit examples.