1 research outputs found
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 where is the cyclic group of order
and is a primitive -th root of unity. The "unknown"
part of the sequence is a group. . splits as and is explicitly known. is a quotient of an in some
sense simpler group . In 1977 Kervaire and Murthy conjectured
that for semi-regular primes , V_n^+ \cong \mathcal{V}_n^+ \cong
\cl^{(p)}(\Q (\zeta_{n-1}))\cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}, where
is the index of regularity of . Under an extra condition on the prime
, Ullom calculated in 1978 in terms of the Iwasawa invariant
as .
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 (where
). 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.Comment: 7 pages, Continuation of NT/020728