Cyclic covers of the projective line, their jacobians and endomorphisms
type: Item
instance of: scholarly article
title: Cyclic covers of the projective line, their jacobians and endomorphisms
zbMATH Open document ID: 0988.14015
publication date: 2002-03-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0008134
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14B99
zbMATH DE Number: 1721429
zbMATH Keywords: cyclic covers of the projective line
zbMATH Keywords: endomorphism ring
zbMATH Keywords: jacobian
author: Yu. G. Zarhin
MaRDI profile type: Publication
OpenAlex ID: W2963458461
arXiv ID: math/0008134
cites work: Q3684278
cites work: Q5340151
cites work: Q4103728
cites work: Q4839675
cites work: Q4090463
cites work: Q5569508
cites work: Q4126536
cites work: Q3926778
DOI: 10.1515/CRLL.2002.030
published in: Journal für die Reine und Angewandte Mathematik
review text: We study the endomorphism ring \(\text{End} (J(C))\) of the complex jacobian \(J(C)\) of a curve \(y^p=f(x)\) where \(p\) is an odd prime and \(f(x)\) is a polynomial with complex coefficients of degree \(n>4\) and without multiple roots. Assume that all the coefficients of \(f\) lie in a (sub)field \(K\) and the Galois group of \(f\) over \(K\) is either the full symmetric group \(S_n\) or the alternating group \(A_n\). Then we prove that \(\text{End} (J(C))\) is the ring of integers \(\mathbb{Z} [\zeta_p]\) in the \(p\)-th cyclotomic field \(\mathbb{Q} [\zeta_p]\) if \(p\) is a Fermat prime (e.g., \(p=3, 5, 17, 257)\). Notice that recently the author extended this result to the case of an arbitrary odd prime \(p\) [\textit{Yu. G. Zarkhin}, ``The endomorphism rings of jacobians of cyclic covers of the projective line'' (\url{http://xxx.lanl.gov/abs/math.AG/0103203})].NEWLINENEWLINENEWLINEThe case of positive characteristic \(\neq p\) (under additional assumptions that \(n>8\) and \(p\mid n)\) is discussed in another one of the author's papers [\textit{Yu. G. Zarkhin}, ``Endomorphism rings of certain jacobians'' (to appear)] there he proves that \(\mathbb{Z} [\zeta_p]\) coincides with its own centralizer in the endomorphism ring of the jacobian.
description: scientific article; zbMATH DE number 1721429