Quadratic points on the Fermat quartic over number fields

TL;DR

This paper proves the finiteness and computability of quadratic points on the Fermat quartic over certain number fields, providing explicit computations for fields of degree less than 8.

Contribution

It establishes conditions under which the quadratic points on the Fermat quartic are finite and explicitly computable, extending understanding of rational points over number fields.

Findings

Finite set of quadratic points on Fermat quartic under specified conditions

Explicit computation of quadratic points for fields with degree less than 8

All quadratic points over odd degree fields are rational over quadratic extensions

Abstract

Let $C$ be a curve defined over a number field $K$. A point $P\in C(\overline{\mathbb{Q}})$ is called $K$-quadratic if $[K(P):K]=2$. Let $K$ be a number field such that the rank of the elliptic curves $E_1:\,y^2= x^3 + 4x$ and $E_2:\,y^2= x^3 - 4x$ over $K$ are $0$. Under the above condition, we prove that the set of $K$-quadratic points on the Fermat quartic $F_4\colon X^4+Y^4=Z^4$ is finite and computable and we provide a procedure to compute this finite set. In particular, we explicitly compute all the $K$-quadratic points if $[K:\mathbb{Q}]<8$. Moreover, if the degree of $K$ is odd, we prove that all the $K$-quadratic points corresponds just to the $\mathbb{Q}$-quadratic points