Comparing the Selmer group of a $p$-adic representation and the Selmer group of the Tate dual of the representation

Abstract

The main conjecture of Iwasawa theory is a conjecture on the relation between a Selmer group and a conjectural $p$-adic $L$-function. This conjectural $p$-adic $L$-function is expected to satisfy a conjectural functional equation in a certain sense. In view of the main conjecture and this conjectural functional equation, one would expect to have certain algebraic relationship between the Selmer group attached to a Galois representation and the Selmer group attached to the Tate twist of the dual of the Galois representation. It is precisely a component of this algebraic relationship that this paper aims to investigate. Namely, for a given "ordinary" $p$-adic representation, we compare its Selmer group with the Selmer group of its Tate dual over an admissible $p$-adic Lie extension, and show that the generalized Iwasawa $\mu$-invariants associated to the Pontryagin dual of the two said Selmer groups are the same. We should mention that in proving the said equality of the $\mu$-invariants, we do not assume the main conjecture nor the conjectural functional equation.