Tokyo Journal of Mathematics

The Hodge Conjecture for The Jacobian Varieties of Generalized Catalan Curves

Noboru AOKI

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Abstract

In this paper we prove that the Hodge conjecture is true for any self-product of the jacobian variety $J(C_{p^\mu,q^\nu})$ of the curve $C_{p^\mu,q^\nu} : y^{q^\nu}=x^{p^\mu}-1$, where $p^\mu$ and $q^\nu$ are powers of distinct prime numbers $p$ and $q$. We also prove that the Hodge ring of $J(C_{p^\mu,q^\nu})$ is {\it not} generated by the divisor classes whenever $p^\mu q^\nu\neq 12$ and $(\mu,\nu)\neq(1,1)$.

Article information

Source
Tokyo J. Math., Volume 27, Number 2 (2004), 313-335.

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208392

Digital Object Identifier
doi:10.3836/tjm/1244208392

Mathematical Reviews number (MathSciNet)
MR2107506

Zentralblatt MATH identifier
1073.14015

Citation

AOKI, Noboru. The Hodge Conjecture for The Jacobian Varieties of Generalized Catalan Curves. Tokyo J. Math. 27 (2004), no. 2, 313--335. doi:10.3836/tjm/1244208392. https://projecteuclid.org/euclid.tjm/1244208392


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