2024/10/23 by Guilherme Dias, Saeed Tafazolian, Jaap Top · 1 citation
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Mathematical functions and polynomials
paper · doi:10.1016/j.ffa.2024.102521
openalex publication_date 2024/10/23 · crossref created 2024/10/23 · crossref deposited 2024/12/05 · crossref issued 2025/01/01 · crossref published 2025/01/01 · crossref published-print 2025/01/01 · openalex created_date 2025/10/10 · crossref indexed 2026/07/29 · openalex updated_date 2026/07/29
This paper studies curves defined using Chebyshev polynomials φ d ( x ) over finite fields. Given the hyperelliptic curve C corresponding to the equation v 2 = φ d ( u ) , the prime powers q ≡ 3 mod 4 are determined such that φ d ( x ) is separable and C is maximal over F q 2 . This extends a result from [30] that treats the special cases 2 | d as well as d a prime number. In particular a proof of [30, Conjecture 1.7] is presented. Moreover, we give a complete description of the pairs ( d , q ) such that the projective closure of the plane curve defined by v d = φ d ( u ) is smooth and maximal over F q 2 . A number of analogous maximality results are discussed.