5 research outputs found
Constructions of diagonal quartic and sextic surfaces with infinitely many rational points
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
with infinitely many rational points and
satisfying the condition . 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 , , , or
, with infinitely many rational points.Comment: revised version will appear in International Journal of Number Theor