Abstract
We compute relations of rational equivalence among special codimension 2 cycles on families of Abelian surfaces using elements of a higher Chow group. These relations are similar to those between Heegner points and special divisors obtained by Zagier, Van der Geer and others.
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Bertolini, M. and Darmon, H.: Heegner points on Mumford-Tate curves, Invent. Math. 126(3) (1996), 413-456.
Besser, A.: CM cycles over Shimura curves, J. Algebraic Geom. 4(4) (1995), 659-691.
Besser, A.: Elliptic fibrations of K3 surfaces and QM Kummer surfaces, Math. Z. 228(2) (1998), 283-308.
Borcherds, R. E.: The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97(2) (1999), 219-233.
Bloch, S.: Algebraic cycles and higher K-theory, Adv. in Math. 61(3) (1986), 267-304.
Collino, A.: Griffiths' infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6(3) (1997), 393-415.
Farkas, H. M. and Kra, I.: Riemann surfaces, Grad. Texts in Math., 71, Springer-Verlag, New York, 1980, xi+337 pp.
Van der Geer, G.: On the geometry of a Siegel modular threefold, Math. Ann. 260(3) (1982), 317-350.
Hirzebruch, F. and Van der Geer, G.: Lectures on Hilbert Modular Surfaces, Based on notes taken by W. Hausmann and F. J. Koll. Seminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], 77. Presses de l'Université de Montréal, Montréal, Que., 1981. 193 pp. ISBN: 2-7606-0562-0.
Gross, B. H.: Heegner points on X 0(N). In Modular Forms (Durham, 1983), Ellis HorwoodSer.Math.Appl.:Statist.Oper.Res., Horwood, Chichester, 1984, pp.87-105.
Gross, B. H. and Zagier, D. B.: Heegner points and derivatives of L-Series, Invent. Math. 84(2) (1986), 225-320.
Hashimoto, K.: Explicit form of quaternion modular embeddings, Osaka J. Math. 32(3) (1995), 533-546.
Humbert, G.: Collected Works, Gauthier-Villars, Paris, 1929.
Igusa, J-I.: On Siegel modular forms of genus 2(II), Amer. J. Math. 86 (1964), 392-412.
Kani, E.: Elliptic curves on Abelian surfaces, Manuscripta Math. 84(2) (1994), 199-223.
Nekovář, J.: Kolyvagin's method for Chow groups of Kuga-Sato varieties, Invent. Math. 107(1) (1992), 99-125.
Mumford, D.: Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970.
Roberts, D. P.: Shimura Curves Analogous to X 0(N), Thesis, Harvard University, May 1989.
Schoen, C.: Complex multiplication cycles and a conjecture of Beilinson and Bloch, Trans. Amer. Math. Soc. 339(1) (1993), 87-115.
Schoen, C.: Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53(3) (1986), 771-794.
Scholl, A. J.: Vanishing cycles and non-classical parabolic cohomology, Invent. Math. 124(1–3) (1996), 503-524.
Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions, Reprint of the 1971 original. Publications of the Mathematical Society of Japan, 11. Kano Memorial Lectures, 1. Princeton University Press, Princeton, NJ, 1994.
Vignéras, M-F.: Arithmétique des algèbres de quaternions (French) [Arithmetic of quaternion algebras] Lecture Notes in Mathematics, 800, Springer, Berlin, 1980.
Weil, A.: Zum Beweis des Torellischen Satzes, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1957 (1957), 33-53 (Collected Works, vol II).
Zagier, D.: Modular Points, Modular Curves, Modular Surfaces and Modular Forms. Workshop, Bonn 1984 (Bonn, 1984), 225-248, Lecture Notes in Math. 1111, Springer, Berlin-New York, 1985.
Zhang, S.: Heights of Heegner cycles and derivatives of L-series. Preprint.
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Sreekantan, R. Relations Among Heegner Cycles on Families of Abelian Surfaces. Compositio Mathematica 127, 243–271 (2001). https://doi.org/10.1023/A:1017515926669
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DOI: https://doi.org/10.1023/A:1017515926669