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

D. Yu. Logachev

Dalʹnevostočnyj matematičeskij žurnal, Tome 9 (2009) no. 1, pp. 105-130

Résumé

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 $H^7$ 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.