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

TL;DR

This paper investigates the algebraic relationship between Selmer groups of a p-adic Galois representation and its Tate dual, showing their Iwasawa μ-invariants are equal without assuming the main conjecture.

Contribution

It establishes the equality of μ-invariants for Selmer groups of a p-adic representation and its Tate dual over p-adic Lie extensions, independently of the main conjecture.

Findings

μ-invariants of the Selmer groups are equal

Comparison holds without assuming the main conjecture

Provides new algebraic insights into Selmer groups and duality

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.