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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 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. This criteria is both of theoretical and computational interest as we show how to use it in practice. We also give refinements and explicit examples.

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 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. This criteria is both of theoretical and computational interest as we show how to use it in practice. We also give refinements and explicit examples.
Paper Structure (2 sections, 3 theorems, 10 equations)

This paper contains 2 sections, 3 theorems, 10 equations.

Table of Contents

  1. A transitive action on the 2-torsion
  2. A non-transitive action on the 2-torsion

Key Result

Proposition 1

Let $C$ be a nice (smooth, projective and geometrically integral) curve defined over a number field $\mathbb{K}$, and let $J$ be its Jacobian. Suppose $C$ has a $\mathbb{K}$-rational divisor class $P_0$ of degree $1$ (e.g. a rational point). Then, at least one of the following statements is true:

Theorems & Definitions (5)

  • Proposition 1
  • proof : First proof
  • proof : Second proof
  • Corollary 2
  • Corollary 3