On Fermat quartics $x^4+y^4=Dz^4$ over cubic fields
Volume 207 / 2023
Abstract
Quartic curves of type $F(x^2,y^2,z^2)=0$ for a homogeneous rational quadratic form $F$ were studied by Cassels (1985), who investigated whether or not they could contain points lying in a cubic number field. Bremner (1986) restricted attention to the ‘Fermat’ type curves $\Gamma _D:x^4+y^4=D z^4$ for a fixed $D \in \mathbb{Z} $. If either of the genus 1 quartics $D_1:X^2+y^4=Dz^4$, $D_3:x^4+y^4=DZ^2$ has no rational points, then $\Gamma _D$ has no points in any cubic field. Suppose the quartics represent elliptic curves. It was shown that if $D \neq 2m^2$ then the rank of $D_1(\mathbb{Q})$ at most 1 implies again $\Gamma _D$ has no points in any cubic field. Here, we complete the ‘missing’ case to show the same result holds when $D=2m^2$. Bremner & Tho (2022) ask: if $\Gamma _D$ is everywhere locally solvable, and ${\rm rank}(D_1(\mathbb{Q} )) \geq 2$, ${\rm rank}(D_2(\mathbb{Q} )) \geq 1$, does $\Gamma _D$ necessarily contain points in some cubic field? This is the case for all such $D$ in the range they consider, $D \lt 10000$. Here, we show that the smallest $D$ providing a counterexample to the question is $D=93586$.
