The arithmetic of a twist of the Fermat quartic

TL;DR

This paper investigates the arithmetic properties of a specific twist of the Fermat quartic, explicitly computing its Jacobian's Mordell--Weil group and revealing its non-trivial Brauer group, with implications for local-global principles.

Contribution

It provides an explicit calculation of the Mordell--Weil group and Picard group of the twisted Fermat quartic, highlighting novel arithmetic phenomena.

Findings

Mordell--Weil group is a free Z/2Z-module of rank 3

Picard group has a free Z/2Z-module of rank 2

The relative Brauer group is non-trivial

Abstract

We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the Mordell--Weil group of the Jacobian variety explicilty. We show that the degree $0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $3$. Thus the relative Brauer group is non-trivial. We also show that this quartic violates the local-global property for linear determinantal representations.