Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

TL;DR

This paper explicitly computes the mod 4 Galois representation for the Fermat quartic's Jacobian, revealing its image as a dihedral group of order 8, and determines rational points over specific fields.

Contribution

It provides the first explicit basis for the 4-torsion points and fully describes the Galois action, completing Faddeev's 1960 analysis.

Findings

Galois image is isomorphic to the dihedral group of order 8

Explicit basis for 4-torsion points on the Jacobian

Complete classification of points over quadratic extensions of the 8th cyclotomic field

Abstract

We use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8-th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic field. Thus we complete Faddeev's work in 1960.