Artin-Schreier curves given by \mathbb F_q-linearized polynomials

2020/12/02 by Oliveira, Daniela, Martínez, F. E. Brochero
#11T06 #12E20 #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2012.01534

Abstract

Let \mathbb F_q be a finite field with q elements, where q is a power of an odd prime p. In this paper we associate circulant matrices and quadratic forms with the Artin-Schreier curve y^q - y= x ⋅ F(x) - λ, where F(x) is a \mathbb F_q-linearized polynomial and λ∈ \mathbb F_q. Our results provide a characterization of the number of affine rational points of this curve in the extension \mathbb F_q^r of \mathbb F_q, for gcd(q,r)=1. In the case F(x) = x^qi-x we give a complete description of the number of affine rational points in terms of Legendre symbols and quadratic characters.

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