Constructions of diagonal quartic and sextic surfaces with infinitely many rational points-Reference-Cited by-同舟云学术

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

Published:2014-09-09 Issue:07 Volume:10 Page:1675-1698
ISSN:1793-0421
Container-title:International Journal of Number Theory
language:en
Short-container-title:Int. J. Number Theory

Author:

Bremner Andrew¹,Choudhry Ajai²,Ulas Maciej³

Affiliation:

1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA

2. 13/4 A Clay Square, Lucknow-226001, India

3. Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Abstract

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.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

Link

https://www.worldscientific.com/doi/pdf/10.1142/S179304211450050X

Reference15 articles.

1. ON xa ± yb ± zc ± wd = 0, 1/a + 1/b + 1/c + 1/d = 1

2. On certain diophantine equations of diagonal type

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