Fine Structure of Class Groups of Prime Power Cyclotomic Fields and the Kervaire-Murthy Conjectures

Abstract

In 1977 Kervaire and Murthy presented three conjectures regarding $K_0 \mathbb{Z} C_{p^n}$, where $C_{p^n}$ is the cyclic group of order $p^n$ and $p$ is a semi-regular prime. The Mayer-Vietoris exact sequence provides the following short exact sequence $$0\to V_n\to \pic (\mathbb{Z} C_{p^n})\to \cl \mathbb{Q} (\zeta_{n-1})\times \pic (\mathbb{Z} C_{p^{n-1}})\to 0$$ where $\zeta_{n-1}$ is a primitive $p^n$-th root of unity. The group $V_n$ that injects into $\pic \mathbb{Z} C_{p^n}\cong\tilde{K}_0\mathbb{Z} C_{p^n}$, is a canonical quotient of an in some sense simpler group $\mathcal{V}_n$. Both groups split in a ``positive'' and ``negative'' part. While $V_n^-$ is well understood there is still no complete information on $V_n^+$. Kervaire and Murthy showed that $K_0 \Z C_{p^n}$ and $V_n$ are tightly connected to class groups of cyclotomic fields. They conjectured that $V_n^+\cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}$, where $r(p)$ is the index of regularity of the prime $p$ and that $\mathcal{V}_n^+\cong V_n^+$, and moreover, $\char\mathcal{V}_n^+\cong \cl^{(p)} \mathbb{Q} (\zeta_{n-1})$, the $p$-part of the class group. In the present paper we calculate $\mathcal{V}_n^+$ and prove that $\char \mathcal{V}_n^+\cong \cl^{(p)} \mathbb{Q}(\zeta_{n-1})$ for all semi-regular primes which also gives us the structure of $\cl^{(p)} \mathbb{Q}(\z_{n-1})$ as an abelian group. Moreover we conclude that all three Kervaire and Murthy conjectures hold is equivalent to that the Iwasawa invariant $\lambda$ equals $r(p)$ and that this also implies that the Iwasawa invariant $\nu$ equals $r(p)$.