Control theorem and functional equation of Selmer groups over $p$-adic Lie extensions

TL;DR

This paper establishes a control theorem and a functional equation for Selmer groups over $p$-adic Lie extensions, extending previous results and exploring their relation to $p$-adic $L$-functions in non-commutative Iwasawa theory.

Contribution

It proves a generalized control theorem and a novel functional equation for Selmer groups in the setting of non-commutative Iwasawa theory, with new proof techniques.

Findings

Control theorem for Selmer groups over $p$-adic Lie extensions.

Functional equation for dual Selmer groups in this setting.

Compatibility with the functional equation of $p$-adic $L$-functions.

Abstract

Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even in the case where the Selmer group is associated to an elliptic curve. We also discuss the functional equation for the analytic $p$-adic $L$-functions and check the compatibility with the functional equation of the dual Selmer groups.