FINE STRUCTURE OF CLASS GROUPS Cl Q(ζn) AND THE KERVAIRE–MURTHY CONJECTURES II OLA HELENIUS AND ALEXANDER STOLIN

There is an Mayer-Vietoris exact sequence 0 → Vn → PicZCpn → Cl Q(ζn−1) × Pic ZCpn−1 → 0 involving the Picard group of the integer group ring ZCpn where Cpn is the cyclic group of order p and ζn−1 is a primitive p -th root of unity. The group Vn splits as Vn ∼= V + n ⊕ V − n and V − n is explicitly known. V + n is a quotient of an in some sense simpler group Vn. In 1977 Kervaire and Murthy conjectured that for semi-regular primes p, V + n ∼= Vn ∼= Cl (Q(ζn−1)) ∼= (Z/p Z), 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 λ as V + n ∼= (Z/pZ) ⊕ (Z/pZ). In the previous paper we proved that for all semi-regular primes, Vn ∼= Cl(Q(ζn−1)) and that these groups are isomorphic to (Z/pZ)0 ⊕ (Z/pZ)10 ⊕ . . .⊕ (Z/pZ)n−1n−2 for a certain sequence {rk} (where r0 = r(p)). Under Ulloms extra condition it was proved that V + n ∼= Vn ∼= Cl(Q(ζn−1)) ∼= (Z/p Z) ⊕ (Z/pZ). 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. The present paper is a continuation of the paper [H-S] of the authors and we refer you there for a more thorough introduction. Let p be an odd prime, Cpn denote the cyclic group of order p n and let ζn be a primitive p-th root of unity. In this paper we work on problems related to Pic(ZCpn). Our methods also lead to the calculation of the p-part of the ideal class group of Z[ζn]. Recall that calculating Picard groups for a group ring like the one above is equivalent to calculating K0 groups. There is a well known exact sequence involving the Picard group of ZCpn which was for example presented by Kervaire and Murthy in [K-M]. The sequence, 1991 Mathematics Subject Classification. 11R65, 11R21, 19A31.