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Complex multiplication cycles on elliptic modular threefolds. (English) Zbl 0623.14018
The author studies one-cycles, modulo algebraic equivalence, on certain elliptic modular threefolds. To be more precise, he considers a desingularization \(\tilde W(N)\) of the fibre product \(Y(N)\times_{\pi}Y(N)\), where \(\pi: Y(N)\to X(N)\) is Shioda’s elliptic modular surface for level N structure, which is viewed as a generalized elliptic curve over the modular curve X(N). The main result is that the \({\mathbb{Q}}\)-vector space \(B_ 1(\tilde W(N))\otimes {\mathbb{Q}}\) of one- cycles, modulo algebraic equivalence, of \(\tilde W(N)\) is infinite dimensional when \(N\geq 3\). In order to establish this he studies certain cycles in the complex multiplication fibres which are homologous to zero, but not algebraically equivalent to zero. Using an idea of Bloch the author is able to show that the images of the cycles are of infinite order. Finally, by choosing an appropriate infinite collection of these cycles, which behave sufficiently well with respect to the action of Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\), he shows that the classes of these cycles are linearly independent in \(B_ 1(\tilde W(N))\otimes {\mathbb{Q}}\).
Reviewer: S.Kamienny

14J30 \(3\)-folds
14C15 (Equivariant) Chow groups and rings; motives
14K22 Complex multiplication and abelian varieties
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Full Text: DOI
[1] A. Ash, et al., Smooth compactification of locally symmetric varieties , Math. Sci. Press, Brookline, Mass., 1975. · Zbl 0334.14007
[2] S. Bloch, Torsion algebraic cycles and a theorem of Roitman , Compositio Math. 39 (1979), no. 1, 107-127. · Zbl 0463.14002 · numdam:CM_1979__39_1_107_0 · eudml:89412
[3] S. Bloch, Algebraic cycles and values of \(L\)-functions , J. Reine Angew. Math. 350 (1984), 94-108. · Zbl 0527.14008 · doi:10.1515/crll.1984.350.94 · crelle:GDZPPN002201496 · eudml:152633
[4] S. Bloch, Algebraic cycles and values of \(L\)-functions. II , Duke Math. J. 52 (1985), no. 2, 379-397. · Zbl 0628.14006 · doi:10.1215/S0012-7094-85-05219-6
[5] H. Clemens, Homological equivalence, modulo algebraic equivalence, is not finitely generated , Inst. Hautes Études Sci. Publ. Math. (1983), no. 58, 19-38 (1984). · Zbl 0529.14002 · doi:10.1007/BF02953771 · numdam:PMIHES_1983__58__19_0 · eudml:103992
[6] D. Cox and S. Zucker, Intersection numbers of sections of elliptic surfaces , Invent. Math. 53 (1979), no. 1, 1-44. · Zbl 0444.14004 · doi:10.1007/BF01403189 · eudml:142653
[7] P. Deligne, Formes modulaires et représentations de \(\mathrm GL(2)\) , Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 55-105. Lecture Notes in Math., Vol. 349. · Zbl 0271.10032
[8] F. El Zein and S. Zucker, Extendability of normal functions associated to algebraic cycles , Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) ed. Philip Griffiths, Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 269-288. JSTOR: · Zbl 0545.14017 · links.jstor.org
[9] A. Fröhlich and J. Queyrut, On the functional equation of the Artin \(L\)-function for characters of real representations , Invent. Math. 20 (1973), 125-138. · Zbl 0256.12010 · doi:10.1007/BF01404061 · eudml:142209
[10] W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[11] S. Lang, Elliptic functions , Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. · Zbl 0316.14001
[12] G. Shimura, Introduction to the arithmetic theory of automorphic functions , Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971. · Zbl 0221.10029
[13] T. Shioda, On elliptic modular surfaces , J. Math. Soc. Japan 24 (1972), 20-59. · Zbl 0226.14013 · doi:10.2969/jmsj/02410020
[14] S. Zucker, Hodge theory with degenerating coefficients. \(L_2\) cohomology in the Poincaré metric , Ann. of Math. (2) 109 (1979), no. 3, 415-476. JSTOR: · Zbl 0446.14002 · doi:10.2307/1971221 · links.jstor.org
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