The ranks of twists of an elliptic curve in characteristic $3$

TL;DR

This paper investigates the Mordell-Weil ranks of twists of a specific elliptic curve over function fields in characteristic 3, relating these ranks to zeta functions and describing elliptic fibrations.

Contribution

It introduces a method to compute ranks of twists of elliptic curves over function fields in characteristic 3, connecting them to zeta functions and fiber structures.

Findings

Ranks expressed via zeta functions of the base curve

Explicit descriptions of fibers in elliptic fibrations

Construction of twists over function fields in characteristic 3

Abstract

Starting from the elliptic curve $E: y^2 = x^3 - x$ over $\mathbb{F}_9$, a curve $\mathcal{X}$ over $\mathbb{F}_{3^{2n}}$ and a cyclic cover of $\mathcal{X}$ of degree $m \in \{2,3,4,6\}$, we construct the corresponding $m$-twists over the function field $\mathbb{F}_{3^{2n}}(\mathcal{X})$. We also obtain the Mordell-Weil rank of these twists in terms of the Zeta functions of $\mathcal{X}$ and of suitable Kummer and Artin-Schreier extensions of it. Finally, we also describe the fibers of the elliptic fibration associated to such twists in the case $\mathcal{X} = \mathbb{P}^1$.