Tate-Shafarevich results for quartic twists in characteristic $2$

TL;DR

This paper investigates elliptic curves over function fields of characteristic 2, providing a rank formula for quartic twists of supersingular elliptic curves and demonstrating the potential for arbitrarily large Mordell-Weil ranks.

Contribution

It introduces a new rank formula for elliptic curves arising as quartic twists in characteristic 2, linking Mordell-Weil rank to the genera of specific curves.

Findings

Explicit rank formula in terms of curve genera

Construction of elliptic curves with arbitrarily large Mordell-Weil rank

Analysis of quartic twists of supersingular elliptic curves

Abstract

The aim of this paper is to present elliptic curves defined over function fields of even characteristic having arbitrarily large Mordell-Weil rank. More precisely, we study elliptic curves arising as quartic twist of a supersingular elliptic curve defined over $\mathbb{F}_2$ using the function field of a maximal curve $C$ that admits an order 4 automorphism. For such elliptic curves we provide a rank formula for its Mordell-Weil group in terms of the genera of $C$ and of another curve covered by $C$.