Far Eastern Mathematical Journal

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Reduction of a problem of finiteness of Tate-Shafarevich group to a result of Zagier type

D. Yu. Logachev

2009, issue 1-2, Ñ. 105–130

Kolyvagin proved that the Tate-Shafarevich group of an elliptic curve over $\mathbb{Q}$ of analytic rank 0 or 1 is finite, and that its algebraic rank is equal to its analytic rank. A program of generalisation of this result to the case of some motives which are quotients of cohomology motives of high-dimensional Shimura varieties and Drinfeld modular varieties is offered. We prove some steps of this program, mainly for quotients of H7 of Siegel sixfolds. For example, we “almost” find analogs of Kolyvagin's trace and reduction relations. Some steps of the present paper are new contribution, because they have no analogs in the case of elliptic curves. There are still a number of large gaps in the program. The most important of these gaps is a high-dimensional analog of a result of Zagier about ratios of Heegner points corresponding to different imaginary quadratic fields on a fixed elliptic curve. The author suggests to the readers to continue these investigations.

Tate-Shafarevich group, motives, Shimura varieties, Euler systems

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[1] V. A. Kolyvagin, “Finiteness of $E(\mathbb{Q})$ and $Sh(E,\mathbb{Q})$ for a subclass of Weil curves”, Math. USSR Izvestiya, 32:3 (1989), 523–541.
[2] V. A. Kolyvagin, “Euler systems”, The Grothendieck Festschrift, 2, Birkhauser, Boston Basel Stuttgart, 1990, 435–483.
[3] D. Logachev, Relations between conjectural eigenvalues of Hecke operators on submotives of Siegel varieties, arXiv: 0405442.
[4] D. Logachev, Action of Hecke correspondences on subvarieties of Shimura varieties, arXiv: 0508204.
[5] D. Logachev, “Action of Hecke correspondences on Heegner curves on a Siegel threefold”, J. of Algebra, 236:1 (2001), 307–348.
[6] Don Zagier, “Modular points, modular curves, modular surfaces and modular forms”, Workshop Bonn 1984 (Bonn, 1984), Lecture Notes in Math., 1111, Springer, Berlin, 1985, 225–248.
[7] B. H. Gross, W. Kohnen, D. B. Zagier, “Heegner points and derivatives of L-series, II”, Math. Ann., 278, 1987.
[8] J. Nekovar, “Kolyvagin's method for Chow groups of Kuga – Sato varieties”, Inv. Math., 107, 1992, 99–125.
[9] B. H. Gross, “Kolyvagin's work on modular elliptic curves”, L-functions and Arithmetic, Proceedings of the Durham Symposium (July, 1989), Cambridge Univ. Press, 1991, 235–256.
[10] P. Deligne, “Travaux de Shimura”, Seminaire Bourbaki 1970/71, Expose 389, Lect. Notes in Math., 244, 1971, 123–165.
[11] V. A. Kolyvagin, D. Yu. Logachev, “Finiteness of Shover totally real fields”, Math. USSR Izvestiya, 39:1 (1992), 829–853.
[12] S. Bloch, L. Kato, “L-functions and Tamagava numbers of motives”, The Grothendieck Festschrift, 1, Birkhauser, Boston Basel Stuttgart, 1990, 333–400.
[13] G. Faltings, Chai Ching-Li, Degeneration of abelian varieties, Springer, 1990.
[14] D. Blasius, J. D. Rogawski, “Zeta functions of Shimura varieties”, Motives. Proc. of Symp. in Pure Math., no. 2, 1994, 525–571 .
[15] A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Mathematical Surveys and Monographs, 67, Second edition, American Mathematical Society, Providence, RI, 2000.

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