Rational points on Cubic, Quartic and Sextic Curves over Finite Fields

2019/12/24 by Oliveira, José Alves
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1912.11441

Abstract

Let \mathbbF_q denote the finite field with q elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over \mathbbF_q in terms of the number of rational points on elliptic curves. In the case where q is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of the forms ax³+by³+cz³=0 and ax⁴+by⁴+cz⁴=0.

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