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

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.

The arithmetic of a twist of the Fermat quartic

“…It turns out that all of them are defined over Q( √ −3) and they are the points of tangency of bitangents defined over Q. (The following result also follows from the results in [13,Section 7].) Theorem 1.4.…”

“…(This subgroup is denoted by D ∞ /F ∞ in [17].) In [13], with the aid of computers, we proved there are no other elements in Jac(F 4 )(Q(ζ 8 )). Thus, we have an isomorphism…”

“…We shall explicitly calculate the Picard group Pic(C 4 ) and the Mordell-Weil group of the Jacobian variety Jac(C 4 ). Theses results rely on our previous calculation of the Mordell-Weil group of the Jacobian variety of the Fermat quartic over Q(ζ 8 ) summarized in [13].…”

“…Via the above isomorphism ρ, we shall translate the results in [13] into the results on divisor classes on C 4 . We define A i , B i , C i ∈ C 4 (0 ≤ i ≤ 3) by…”

“…Here is a brief sketch of the proof of our results. In [13], based on the results of Faddeev and Rohrlich, we explicitly calculated the Mordell-Weil group of the Fermat quartic…”