Homotopy groups of spectra and $p$-adic $L$-functions over totally real number fields

TL;DR

This paper explores the interplay between Euler characteristics in Iwasawa theory over totally real fields and homotopy-theoretic invariants like $K$-theory and Topological Cyclic homology, offering new perspectives on their interactions.

Contribution

It introduces different approaches to understanding Euler characteristics in Iwasawa theory and connects these with homotopy invariants such as $K(1)$-local $K$-theory and Topological Cyclic homology.

Findings

Different methods to analyze Euler characteristics in Iwasawa theory.

Connections established between Iwasawa invariants and homotopy-theoretic invariants.

Insights into the interaction of algebraic and topological invariants over totally real fields.

Abstract

The goal of this paper is to illustrate different approaches to understand Euler characteristics in the setting of totally real commutative and non-commutative Iwasawa theory. In addition to this, and in the spirit of Hesselholt and Mitchell, we show how these objects interact with homotopy theoretic invariants such as ($K(1)$-local) $K$-theory and Topological Cyclic homology.