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

Abstract

We use explicit methods to study the [Formula: see text]-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 [Formula: see text]-torsion points. We calculate the Galois action, and show that the image of the mod [Formula: see text] Galois representation is isomorphic to the dihedral group of order [Formula: see text]. As applications, we calculate the Mordell–Weil group of the Jacobian variety of the Fermat quartic over each subfield of the [Formula: see text]th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the [Formula: see text]th cyclotomic field. Thus, we complete Faddeev’s work in 1960.