Rational points on Fermat curves over finite fields

2017 ◽  
Vol 16 (03) ◽  
pp. 1750046
Author(s):  
Wei Cao ◽  
Shanmeng Han ◽  
Ruyun Wang
Keyword(s):  
Finite Fields ◽  
Rational Point ◽  
Rational Points ◽  
Fermat Curve ◽  
Fermat Curves ◽  
Curves Over Finite Fields

Let [Formula: see text] be the [Formula: see text]-rational point on the Fermat curve [Formula: see text] with [Formula: see text]. It has recently been proved that if [Formula: see text] then each [Formula: see text] is a cube in [Formula: see text]. It is natural to wonder whether there is a generalization to [Formula: see text]. In this paper, we show that the result cannot be extended to [Formula: see text] in general and conjecture that each [Formula: see text] is a cube in [Formula: see text] if and only if [Formula: see text].

ON THE NUMBER OF RATIONAL POINTS OF GENERALIZED FERMAT CURVES OVER FINITE FIELDS

2012 ◽  
Vol 08 (04) ◽  
pp. 1087-1097 ◽  
Author(s):  
STEFANIA FANALI ◽  
MASSIMO GIULIETTI
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Direct Product ◽  
Finite Fields ◽  
Classical Problem ◽  
Rational Points ◽  
Fermat Curve ◽  
Cyclic Groups ◽  
Fermat Curves ◽  
Curves Over Finite Fields

The Stöhr–Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C1 and C2 of the same size k, and such that the quotient curves with respect to both C1 and C2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves.

scholarly journals RATIONAL POINTS ON SOME FERMAT CURVES AND SURFACES OVER FINITE FIELDS

2014 ◽  
Vol 10 (02) ◽  
pp. 319-325 ◽  
Author(s):  
JOSÉ FELIPE VOLOCH ◽  
MICHAEL E. ZIEVE
Keyword(s):  
Finite Fields ◽  
Rational Points ◽  
Explicit Description ◽  
Fermat Curve ◽  
Curves And Surfaces ◽  
Fermat Curves

We give an explicit description of the 𝔽qi-rational points on the Fermat curve uq-1 + vq-1 + wq-1 = 0, for i ∈{1, 2, 3}. As a consequence, we observe that for any such point (u, v, w), the product uvw is a cube in 𝔽qi. We also describe the 𝔽q2-rational points on the Fermat surface uq-1 + vq-1 + wq-1 + xq-1 = 0, and show that the product of the coordinates of any such point is a square.

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