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 -invariants of the fine Selmer groups and the mu -invariants of the Selmer groups to be bounded as one runs over the mathbb {Z}_p-extensions of K. Moreover, we describe a criterion for the boundedness of Iwasawa lambda -invariants of Selmer groups and fine Selmer groups over multiple mathbb {Z}_p-extensions which generalises a criterion of Monsky from dimension 2 to arbitrary dimension.