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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
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[1] N. Aoki, On some arithmetic problems related to the Hodge cycles on the Fermat varieties, Math. Ann. 266 (1983), 23-54 (Erratum: Math. Ann. 267 (1984), p. 572). · Zbl 0506.14030 · doi:10.1007/BF01458703 · eudml:163839
[2] N. Aoki, Some new algebraic cycles on Fermat varieties, J. Math. Soc. Japan 39 (1987), 385-396. · Zbl 0615.14002 · doi:10.2969/jmsj/03930385
[3] N. Aoki, Simple factors of the jacobian of a Fermat curve and the Picard number of a product of Fermat curves, Amer. J. Math. 113 (1991), 779-833. · Zbl 0752.14021 · doi:10.2307/2374786
[4] N. Aoki, Some remarks on the Hodge conjecture for abelian varieties of Fermat type, Comm. Math. Univ. Sancti Pauli 49 (2000), 177-194. · Zbl 1016.14004
[5] N. Aoki, Hodge cycles on CM abelian varieties of Fermat type, Comm. Math. Univ. Sancti Pauli 51 (2002), 99-130. · Zbl 1027.14003
[6] F. Hazama, Hodge cycles on Abelian varieties of CM-type, Res. Act. Fac. Eng. Tokyo Denki Univ. 5 (1983), 31-33.
[7] F. Hazama, Algebraic cycles on certain abelian varieties and powers of special surfaces, J. Fac. Sci. Univ. Tokyo 31 (1985), 487-520. · Zbl 0591.14006
[8] F. Hazama, Algebraic cycles on nonsimple abelian varieties, Duke Math. J. 58 (1989), 31-37. · Zbl 0697.14028 · doi:10.1215/S0012-7094-89-05803-1
[9] F. Hazama, Hodge cycles on the jacobian variety of the Catalan curve, Compostio Math. 107 (1997), 339-353. · Zbl 1044.11587 · doi:10.1023/A:1000106427229
[10] H. Imai, On the Hodge groups of some Abelian varieties, Kodai Math. Sem. Rep. 27 (1976), 367-372. · Zbl 0328.14015 · doi:10.2996/kmj/1138847263
[11] D. S. Kubert and S. Lang, Modular Units , Grundlehren der mathematischen Wissenschaften 244 (1981), Springer. · Zbl 0492.12002
[12] T. Kubota, On the field extension by complex multiplication, Trans. Amer. Math. Soc. 118 (1965), 113-122. · Zbl 0146.27902 · doi:10.2307/1993947
[13] J. D. Lewis, A Survey of the Hodge Conjecture , CRM Monograph series 10 (1991), Amer. Math. Soc. · Zbl 0778.14002
[14] V. K. Murty, Computing the Hodge group of an abelian variety, Séminaire de Théorie des Nombres, Paris 1988-1989, Prog. Math. 91 (1990), Birlhäuser, 141-158. · Zbl 0761.14014
[15] Z. Ran, Cycles on Fermat hypersurfaces, Compositio Math. 42 (1981), 121-142.
[16] K. A. Ribet, Division fields of abelian varieties with complex multiplication, Soc. Math. France Mémorie 2 (1980), 75-94. · Zbl 0452.14009 · numdam:MSMF_1980_2_2__75_0 · eudml:94824
[17] C.-G. Schmidt, Zur Arithmetik abelscher Varietäten mit komplexer Multiplikation , Springer Lect. Notes in Math. 1082 (1984), Springer. · Zbl 0567.14022
[18] C. Schoen, Hodge classes on self-products of a variety with an automorphism, Compositio Math. 65 (1988), 3-32. · Zbl 0663.14006 · numdam:CM_1988__65_1_3_0 · eudml:89882
[19] G. Shimura and Y. Taniyama, Complex multiplication of abelian varieties and its applications to number theory , Math. Soc. Japan 1961. · Zbl 0112.03502
[20] T. Shioda, The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), 175-184. · Zbl 0403.14007 · doi:10.1007/BF01428804 · eudml:163317
[21] T. Shioda, Algebraic cycles on abelian varieties of Fermat type, Math. Ann. 258 (1981), 65-80. · Zbl 0515.14005 · doi:10.1007/BF01450347 · eudml:163579
[22] T. Shioda, What is known about the Hodge conjecture?, Adv. St. in Pure Math. 1 (1983), 55-68. · Zbl 0527.14010
[23] T. Shioda and T. Katsura, On Fermat varieties, Tôhoku Math. J. 31 (1979), 97-115. · Zbl 0415.14022 · doi:10.2748/tmj/1178229881
[24] B. van Geemen, Theta functions and cycles on some Abelian four folds, Math. Z. 221 (1996), 617-631. · Zbl 0862.14029 · doi:10.1007/BF02622136 · eudml:174870
[25] L. C. Washington, Introduction to Cyclotomic Fields , Graduate Texts in Mathematics 83 (1982) Springer. · Zbl 0484.12001
[26] S. P. White, Sporadic cycles on CM abelian varieties, Compositio Math. 88 (1993), 123-142. · Zbl 0798.14025 · numdam:CM_1993__88_2_123_0 · eudml:90241
[27] K. Yamamoto, The gap group of multiplicative relationship of Gauss sums, Symp. Math. XV (1975), 427-440. · Zbl 0319.10044
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