Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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 ranks of the elliptic curves $E_1:\, y^2= x^3 + 4x$ and $E_2:\, y^2= x^3 - 4x$ over $K$ are $0$. Under this 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 it. In particular, we explicitly compute all the $K$-quadratic points if $[K:\mathbb Q] \lt 8$. Moreover, if the degree of $K$ is odd, we prove that the $K$-quadratic points are just the $\mathbb Q$-quadratic points.