Quadratic points on the Fermat quartic over number fields

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 $F_4:X^4+Y^4=Z^4$ 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

To appear in Acta Arithmetica