## Tokyo Journal of Mathematics

### The Hodge Conjecture for The Jacobian Varieties of Generalized Catalan Curves

Noboru AOKI

#### 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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