A trick to ensure positive Mordell–Weil rank
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, this implies that a generic nice curve of genus 𝑔 > 1 with a rational degree-1 class has strictly positive rank.This criteria is of both 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.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Advanced Algebra and Geometry
Analytic Number Theory Research
Publication details
- Journal / source
- Polynesian Journal of Mathematics
- Volume
- 4
- Issue
- 15
- Pages
- 1–14
- Publication date
- 28 September 2026
- Language
- EN