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

abstract: In this paper, we construct several infinite families of diagonal quartic surfaces ax[superscript 4] + by[superscript 4] + cz[superscript 4] + dw[superscript 4] = 0 (where a, b, c, d are non-zero integers) with infinitely many rational points and satisfying the condition abcd is not a 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 ax[superscript 6] + by[superscript 6] + cz[superscript 6] + dw[superscript i] = 0, i = 2, 3, or 6, with infinitely many rational points.Electronic version of an article published in INTERNATIONAL JOURNAL OF NUMBER THEORY, 10, 7, 2014, 1675-1698 Copyright World Scientific Publishing Company http://dx.doi.org/10.1142/S179304211450050