The Fermat curve of fourth degree
type: Item
instance of: scholarly article
title: The Fermat curve of fourth degree
zbMATH Open document ID: 0793.14018
DOI: 10.1016/0021-8693(92)90117-5
author: Chong Hai Lim
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: Let \(F_ 4:X^ 4+Y^ 4+Z^ 4=0\) be the Fermat curve of degree 4 over \(\mathbb{Q}\). Each automorphism of the curve \(F_ 4\) induces an endomorphism of the corresponding jacobian variety \(J_ 4\). This defines a map \(\mathbb{Q} [\text{Aut} (F_ 4)] \to \text{End} (J_ 4) \bigotimes \mathbb{Q}\). The author shows by explicit calculations that \(\text{End} (J_ 4)\) is precisely the image of the group ring \(\mathbb{Z} [\text{Aut} (J_ 4)]\). It is then a consequence that \(J_ 4\) is not isomorphic to a product of elliptic curves.
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 165840
zbMATH Keywords: automorphism of Fermat curve
zbMATH Keywords: endomorphism of jacobian
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90117-5
OpenAlex ID: W1976365929
cites work: Q3931542
cites work: Q3984269
cites work: The geometry of the Jacobian of the Fermat curve of exponent five
cites work: Q3849064
cites work: Q3710637
description: scientific article; zbMATH DE number 165840