Polynesian Journal of Mathematics

Volume 4, Issue 15
Special volume AJAX in honor of René Schoof

A trick to ensure positive Mordell–Weil rank

Thibaut Misme

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.

Citation:

Thibaut Misme. A trick to ensure positive Mordell–Weil rank. Polynesian Journal of Mathematics, Volume 4, Issue 15, Pages 1–14. (Sept. 2026) DOI: 10.69763/polyjmath.4.15

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Milestones:

Received 13 Mar 2026
Revised 19 Aug 2026
Accepted 1 Sep 2026
Published 28 Sep 2026
Communicated by Valerio Talamanca

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