Iwasawa Invariants of Even $K$-groups of Rings of Integers in the $\mathbb{Z}_2$-extension over Real Quadratic Number Fields

TL;DR

This paper investigates the Iwasawa invariants of even K-groups of rings of integers in the cyclotomic $Z_2$-extension over real quadratic fields, providing asymptotic formulas and explicit invariants for various cases.

Contribution

It introduces a new asymptotic formula for the 2-primary parts of K-groups in the cyclotomic extension and explicitly determines Iwasawa invariants for certain real quadratic fields.

Findings

Derived asymptotic formula for 2-primary parts of K-groups

Determined $ ext{lambda}$ and $ ext{mu}$ invariants for large n

Established lower bounds for the validity of the asymptotic formula

Abstract

Let $F$ be a real quadratic number field, and let $F_{cyc}$ denote its cyclotomic $\mathbb{Z}_2$-extension. For each integer $n\geq0$, let $F_n$ be the unique intermediate field in $F_{cyc}$ such that $[F_n:F]=2^n$. By studying the $2$-adic divisibility of Dirichlet $L$-series at negative integers, we derive an asymptotic formula that determines the order of the $2$-primary part of even $K$-groups of rings of integers of $F_n$ for sufficiently large $n$. As a corollary, we determine their $\lambda$ and $\mu$ invariants. We also establish a lower bound for $n$ beyond which this asymptotic formula holds. Our results have two main applications: (1) For $K=\mathbb{Q}$, $\mathbb{Q}(\sqrt{p})$ or $\mathbb{Q}(\sqrt{2p})$ with $p\equiv\pm3\mod 8$, we determine the structure of the $2$-primary tame kernels $K_2\mathcal{O}_{K_n}(2)$; (2) We explicitly determine the three Iwasawa invariants $\lambda,\mu,\nu$ for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors.