FINE STRUCTURE OF CLASS GROUPS Cl Q(ζn) AND THE KERVAIRE-MURTHY CONJECTURES

In 1977 Kervaire and Murthy presented three conjectures regarding K0ZCpn , where Cpn is the cyclic group of order p n and p is a semi-regular prime that is p does not divide h (regular p does not divide the class number h = hh). The Mayer-Vietoris exact sequence provides the following short exact sequence 0 → Vn → PicZCpn → ClQ(ζn−1)× PicZCpn−1 → 0 Here ζn−1 is a primitive p -th root of unity. The group Vn that injects into PicZCpn ∼= K̃0ZCpn , is a canonical quotient of an in some sense simpler group Vn. 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 K0ZCpn and Vn are tightly connected to class groups of cyclotomic fields. They also conjectured that V + n ∼= (Z/pZ), where r(p) is the index of regularity of the prime p and that Vn ∼= V + n , and moreover, CharVn ∼= Cl Q(ζn−1), the p-part of the class group. Under an extra assumption on the prime p, Ullom proved in 1978 that V + n ∼= (Z/pnZ)r(p)⊕(Z/pn−1Z)λ−r(p), where λ is one of the Iwasawa invariants. Hence Kervaire and Murthys first conjecture holds only when λ = r(p). In the present paper we calculate Vn and prove that CharV + n ∼= Cl Q(ζn−1) for all semi-regular primes which also gives us the structure of Cl Q(ζn−1) as an abelian group. We also prove that under the same condition Ullom used, conjecture two always holds, that is Vn ∼= V + n . Under the assumption λ = r(p) we construct a special basis for a ring closely related to ZCpn , consisting of units from a number field. This basis is used to prove that Vn ∼= V + n in this case and it also follows that the Iwasawa invariant ν equals r(p). Moreover we conclude that λ = r(p) is equivalent to that all three Kervaire and Murthy conjectures hold.