Mathematics – Number Theory
Scientific paper
2004-11-15
Mathematics
Number Theory
32 pages; correction of minor errors
Scientific paper
Kolyvagin proved that the Tate-Shafarevich group of an elliptic curve over 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 Shimura varieties is offered. We prove some steps of this program, mainly for quotients of $H^7$ 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.
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