Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fine Structure of Class Groups $\cl^{(p)}\Q(\z_n)$ and the
  Kervaire--Murthy Conjectures II   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.

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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: atom-ph