Dependent rational points on curves over finite fields - Lefschetz theorems and exponential sums

2001 ◽  
Vol 6 ◽  
pp. 297-309 ◽  
Author(s):  
Johan P. Hansen
Keyword(s):  
Finite Fields ◽  
Exponential Sums ◽  
Rational Points ◽  
Curves Over Finite Fields ◽  
Lefschetz Theorems

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].

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