Plane sections of Fermat surfaces over finite fields

Author(s)

Borges, Herivelto ^[1] ; Cook, Gary ^[1] ; Coutinho, Mariana ^[1]

Affiliation

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Ave Trabalhador Sao Carlense 400, BR-13566590 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: FINITE FIELDS AND THEIR APPLICATIONS; v. 52, p. 156-173, JUL 2018.
• Web of Science Citations: 0

Abstract

In this paper, we characterize all curves over F-q arising from a plane section P : X-3 - e(0)X(0) - e(1)X(1) - e(2)X(2) = 0 of the Fermat surface S : x(0)(d) + X-1(d) + X-2(d) + X-3(d) = 0, where q = p(h) = 2d+1 is a prime power, p > 3, and e(0), e(1), e(2) is an element of F-q. In particular, we prove that any nonlinear component G subset of P boolean AND S is a smooth classical curve of degree n <= d attaining the Stohr Voloch bound \#G(F-q) <= 1/2n(n+ q - 1) - 1/2i(n - 2), with i is an element of [0,1,2,3, n, 3n]. (C) 2018 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 17/04681-3 - Algebraic curves over finite fields
• Grantee: Herivelto Martins Borges Filho
• Support Opportunities: Regular Research Grants