Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

Abstract

In this note we construct several infinite families of diagonal quartic surfaces \begin{equation*} ax^4+by^4+cz^4+dw^4=0, \end{equation*} where $a,b,c,d\in \mathbb{Z}\setminus \{0\}$ with infinitely many rational points and satisfying the condition $abcd\ne \square$. In particular, we present an infinite family of diagonal quartic surfaces defined over $\Q$ with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type $a{x}^6+b{y}^6+c{z}^6+d{w}^i=0$, $i=2$, $3$, or $6$, with infinitely many rational points.

revised version will appear in International Journal of Number Theory