The Hodge conjecture for the Jacobian varieties of generalized Catalan curves
type: Item
instance of: scholarly article
title: The Hodge conjecture for the Jacobian varieties of generalized Catalan curves
zbMATH Open document ID: 1073.14015
author: Noboru Aoki
published in: Tokyo Journal of Mathematics
publication date: 2005-03-14T00:00:00Z
review text: The main objective of this article is to show the validity of the Hodge conjecture for the Jacobian variety of the curve \(C_{p^\mu,q^\nu}: y^{p^\nu}= x^{p^\mu}-1\), called a generalized Catalan curve, for any pairs \(p^\mu\), \(q^\nu\) of powers of distinct primes. When \(\mu= \nu= 1\), the case with one of \(p\), \(q\) equal to two is treated by \textit{T. Kubota} [Trans. Am. Math. Soc. 118, 113--122 (1965; Zbl 0146.27902)], and the remaining cases by the reviewer [Compos. Math. 107, No. 3, 341--353 (1997; Zbl 1044.11587)]. These two articles prove the Hodge conjecture by showing that the respective Hodge ring is generated by divisor classes. When \((\mu,\nu)\neq(1,1)\), the Hodge ring of \(J(C_{p^\mu, q^\nu})\) is not generated by divisor classes, and hence the author needs to construct algebraic cycles without the help of divisors. He overcomes this difficulty by noting that an appropriate Fermat curve covers the generalized Catalan curve, and by employing the inductive structure of Fermat varieties.
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 2145294
zbMATH Keywords: Hodge conjecture
zbMATH Keywords: complex multiplication
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208392
OpenAlex ID: W2044267017
Wikidata QID: Q122913722
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DOI: 10.3836/TJM/1244208392
description: scientific article; zbMATH DE number 2145294