Fine Structure of Class Groups $\cl^{(p)}\Q(\z_n)$ and the Kervaire--Murthy Conjectures II

Abstract

There is an Mayer-Vietoris exact sequence involving the Picard group of the integer group ring $\Z C_{p^n}$ where $C_{p^n}$ is the cyclic group of order $p^n$ and $\zeta_{n-1}$ is a primitive $p^n$-th root of unity. The "unknown" part of the sequence is a group. $V_n$. $V_n$ splits as $V_n\cong V_n^+\oplus V_n^-$ and $V_n^-$ is explicitly known. $V_n^+$ is a quotient of an in some sense simpler group $\mathcal{V}_n$. In 1977 Kervaire and Murthy conjectured that for semi-regular primes $p$, $V_n^+ \cong \mathcal{V}_n^+ \cong \cl^{(p)}(\Q (\zeta_{n-1}))\cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}$, where $r(p)$ is the index of regularity of $p$. Under an extra condition on the prime $p$, Ullom calculated $V_n^+$ in 1978 in terms of the Iwasawa invariant $\lambda$ as $V_n^+ \cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}\oplus (\mathbb{Z}/p^{n-1} \mathbb{Z})^{\lambda-r(p)}$. In the previous paper we proved that for all semi-regular primes, $\mathcal{V}_n^+\cong \cl^{(p)}(\Q (\zeta_{n-1}))$ and that these groups are isomorphic to \[(\mathbb{Z}/p^n \mathbb{Z})^{r_0}\oplus (\mathbb{Z}/p^{n-1} \mathbb{Z})^{r_1-r_0} \oplus \hdots \oplus (\mathbb{Z}/p \mathbb{Z})^{r_{n-1}-r_{n-2}} \] for a certain sequence $\{r_k\}$ (where $r_0=r(p)$). Under Ulloms extra condition it was proved that \[V_n^+ \cong \mathcal{V}_n^+ \cong \cl^{(p)}(\Q(\z_{n-1})) \cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}\oplus (\mathbb{Z}/p^{n-1}\mathbb{Z})^{\lambda-r(p)}.\] In the present paper we prove that Ullom's extra condition is valid for all semi-regular primes and it is hence shown that the above result holds for all semi-regular primes.