On the CM character of the curves \(y^ 2=x^ q-1\)
type: Item
instance of: scholarly article
title: On the CM character of the curves \(y^ 2=x^ q-1\)
zbMATH Open document ID: 0882.11035
author: Paul van Wamelen
published in: Journal of Number Theory
publication date: 1997-06-08T00:00:00Z
review text: The author gives a formula for the number of points on the hyperelliptic curve \[ y^2=x^q-1 \] over a field of \(p\) elements. This is done by determining the CM character of the curve and is similar to the method of Gauss in the elliptic curve case of \(q=3\). The resulting algorithm depends on decomposing the prime \(p\) in the \(q\)th cyclotomic field. The author points out that this can be done in probabilistic polynomial time or in deterministic polynomial time if one is willing to assume the Extended Riemann Hypothesis.
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 1017566
zbMATH Keywords: Jacobians
zbMATH Keywords: curves
zbMATH Keywords: CM character
zbMATH Keywords: number of points on the hyperelliptic curve
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: jnth.1997.2089
OpenAlex ID: W1585262649
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DOI: 10.1006/JNTH.1997.2089
description: scientific article; zbMATH DE number 1017566