On the structure of Selmer groups

Abstract

The paper contains some applications of explicit cohomology classes (which the author has constructed earlier using Heegner points) to the theory of Selmer groups of a modular elliptic curve. Moreover, some generalizations of Selmer groups are considered. The case when the Heegner point over the imaginary quadratic field has infinite order was studied in the work [1]. In fact, the theory of [1] is valid under a more general assumption which is, hypothetically, always true and discussed below. For the convenience of the reader, we recall in part 1 the definitions of the Selmer groups and of our explicit cohomology classes, and formulate some of our results. The second part is essentially based on the work [1] and requires some familiarity with it. The second part contains proofs of results for ` ∈ B(E) (see below for notations), formulations of corresponding results for ` 6∈ B(E), and some global consequences of these results.