We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

我们研究了在 固定数域K 的 ${{Z}}_p$ -extensions 中在p处具有良好普通减少的阿贝尔变种的p 主Selmer 群的生长。证明在许多情况下，在K上的 ${{Z}}_p$ 扩展的足够大量有限层中的 Selmer 群的知识 足以限制整体增长，我们将 Selmer 群的 Iwasawa 不变量联系起来在K 的不同 ${{Z}}_p$ -extensions 中。作为应用程序，我们限制了 Mordell-Weil 等级的增长和 Tate-Shafarevich 集团的增长。最后，我们得出了关于精细 Selmer 群增长的类似结果。