Boundedness of Iwasawa Invariants of Fine Selmer Groups and Selmer Groups

Abstract

In this article we consider Selmer groups and fine Selmer groups of abelian varieties over a number field K. Following a classical approach of Monsky for Iwasawa modules from ideal class groups, we give sufficient conditions for the Iwasawa $$\mu $$ μ -invariants of the fine Selmer groups and the $$\mu $$ μ -invariants of the Selmer groups to be bounded as one runs over the $$\mathbb {Z}_p$$ Z p -extensions of K. Moreover, we describe a criterion for the boundedness of Iwasawa $$\lambda $$ λ -invariants of Selmer groups and fine Selmer groups over multiple $$\mathbb {Z}_p$$ Z p -extensions which generalises a criterion of Monsky from dimension 2 to arbitrary dimension.