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

Abstract

Let $F$ be a real quadratic number field, and let $F_{cyc}$ denote its cyclotomic ${\mathbb{Z}}_2$-extension. For each integer $n\ge 0$, 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 $p\equiv \pm 3\mathrm{mod}8$, 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.