TY - JOUR
T1 - Constructions of diagonal quartic and sextic surfaces with infinitely many rational points
AU - Bremner, Andrew
AU - Choudhry, Ajai
AU - Ulas, Maciej
N1 - Publisher Copyright:
© 2014 World Scientific Publishing Company.
PY - 2014/11/16
Y1 - 2014/11/16
N2 - In this paper, we construct several infinite families of diagonal quartic surfaces ax4 + by4 + cz4 + dw4 = 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 ax6 + by6 + cz6 + dwi = 0, i = 2, 3, or 6, with infinitely many rational points.
AB - In this paper, we construct several infinite families of diagonal quartic surfaces ax4 + by4 + cz4 + dw4 = 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 ax6 + by6 + cz6 + dwi = 0, i = 2, 3, or 6, with infinitely many rational points.
KW - Diagonal quartic surface
KW - sextic surface
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U2 - 10.1142/S179304211450050X
DO - 10.1142/S179304211450050X
M3 - Article
AN - SCOPUS:84929518154
SN - 1793-0421
VL - 10
SP - 1675
EP - 1698
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 7
ER -