On Universal Norms for $p$-adic Representations in Higher Rank Iwasawa Theory

TL;DR

This paper develops a systematic theory of universal norms for $p$-adic representations in higher rank Iwasawa theory, introducing new pairings and refining classical conjectures related to Iwasawa invariants and $L$-functions.

Contribution

It constructs a foundational module of higher rank universal norms and an Iwasawa-theoretic pairing, advancing the understanding of $p$-adic representations in Iwasawa theory.

Findings

Refined the classical Iwasawa Main Conjecture for cyclotomic fields.

Established properties of the module of higher rank universal norms.

Applied results to conjectures on Iwasawa invariants and $L$-function leading terms.

Abstract

We begin a systematic investigation of universal norms for $p$-adic representations in higher rank Iwasawa theory. After establishing the basic properties of the module of higher rank universal norms we construct an Iwasawa-theoretic pairing that is relevant to this setting. This allows us, for example, to refine the classical Iwasawa Main Conjecture for cyclotomic fields, and also to give applications to various well-known conjectures in arithmetic concerning Iwasawa invariants and leading terms of $L$-functions.