Rational points on some Fermat curves and surfaces over finite fields
Number Theory
2016-03-04 v1 Algebraic Geometry
Abstract
We give an explicit description of the F_{q^i}-rational points on the Fermat curve u^{q-1}+v^{q-1}+w^{q-1}=0 for each i=1,2,3. As a consequence, we observe that for any such point (u,v,w), the product uvw is a cube in F_{q^i}. We also describe the F_{q^2}-rational points on the Fermat surface u^{q-1}+v^{q-1}+w^{q-1}+x^{q-1}=0, and show that the product of the coordinates of any such points is a square.
Cite
@article{arxiv.1310.1772,
title = {Rational points on some Fermat curves and surfaces over finite fields},
author = {Jose Felipe Voloch and Michael E. Zieve},
journal= {arXiv preprint arXiv:1310.1772},
year = {2016}
}
Comments
8 pages