Characteristic polynomials of the curve \(v^2=u^p-au-b\) over finite fields of characteristic \(p\) (Q1946683)

Hyperelliptic curves over finite fields are interesting objects in arithmetic geometry. There are some applications to coding theory, and low genus hyperelliptic curves are relevant in cryptography. One of the fundamental problems is the computation of the order of the Jacobian of a hyperelliptic curve over an arbitrary finite ground field. One effective method is the \(L(1)\) computation which requires knowledge of the coefficient of the \(L\)-polynomial resp. the computation of the zeta functions of the characteristic polynomials of the given hyperelliptic curve. In this article, the authors consider the specific hyperelliptic curve \(C\) given by the equation \(v^2=u^p-au-b\) over a finite field \(\mathbb{F}_q\) with \(q=p^s\) elements, where \(s\) is a positive integer. The authors extend well-known results of \textit{I. Duursma} and \textit{K. Sakurai} [Proceedings of an international conference on coding theory, cryptography, and related areas, Guanajuato, Mexico, Springer, 73--89 (2000; Zbl 1009.11047)] for \(s=1\) and \textit{L. You} et. al., [Math. Probl. Eng. 2011, 1--25 (2011; Zbl 1213.94144)] for \(s=2\) to determine the characteristic polynomials of the hyperelliptic curve \(C\) for arbitrary \(s\) and \(a,b\in\mathbb{F}_q\). They also provide numerical data for Jacobian group orders with large prime factors for \(p=5,7\).