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.
Details
- Constructions of Diagonal Quartic and Sextic Surfaces With Infinitely Many Rational Points
- Bremner, Andrew (Author)
- Choudhry, Ajai (Author)
- Ulas, Maciej (Author)
- College of Liberal Arts and Sciences (Contributor)
- Digital object identifier: 10.1142/S179304211450050X
- Identifier TypeInternational standard serial numberIdentifier Value1793-0421
- Identifier TypeInternational standard serial numberIdentifier Value1793-7310
- 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/S179304211450050X, opens in a new window
Citation and reuse
Cite this item
This is a suggested citation. Consult the appropriate style guide for specific citation guidelines.
Bremner, Andrew, Choudhry, Ajai, & Ulas, Maciej (2014). Constructions of diagonal quartic and sextic surfaces with infinitely many rational points. INTERNATIONAL JOURNAL OF NUMBER THEORY, 10(7), 1675-1698. http://dx.doi.org/10.1142/S179304211450050X