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The Hodge conjecture for the Jacobian varieties of generalized Catalan curves. (English) Zbl 1073.14015
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 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.
##### MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14K22 Complex multiplication and abelian varieties 14H40 Jacobians, Prym varieties
##### Keywords:
Hodge conjecture; complex multiplication
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##### References:
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