Fine Structure of Class Groups of Prime Power Cyclotomic Fields and the Kervaire-Murthy Conjectures

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In 1977 Kervaire and Murthy presented three conjectures regarding K₀ Z Cₚₙ, where Cₚₙ is the cyclic group of order pⁿ and p is a semi-regular prime. The Mayer-Vietoris exact sequence provides the following short exact sequence 0→ Vₙ→ (Z Cₚₙ)→ Q (ζₙ₋₁)× (Z Cₚⁿ⁻¹)→ 0 where ζₙ₋₁ is a primitive pⁿ-th root of unity. The group Vₙ that injects into Z CₚₙK₀Z Cₚₙ, is a canonical quotient of an in some sense simpler group Vₙ. Both groups split in a ``positive'' and ``negative'' part. While Vₙ- is well understood there is still no complete information on Vₙ+. Kervaire and Murthy showed that K₀ Cₚₙ and Vₙ are tightly connected to class groups of cyclotomic fields. They conjectured that Vₙ+ (Z/pⁿ Z)ʳ⁽ᵖ⁾, where r(p) is the index of regularity of the prime p and that Vₙ+ Vₙ+, and moreover, ₙ+ ⁽ᵖ⁾ Q (ζₙ₋₁), the p-part of the class group. In the present paper we calculate Vₙ+ and prove that Vₙ+ ⁽ᵖ⁾ Q(ζₙ₋₁) for all semi-regular primes which also gives us the structure of ⁽ᵖ⁾ Q(ₙ₋₁) as an abelian group. Moreover we conclude that all three Kervaire and Murthy conjectures hold is equivalent to that the Iwasawa invariant λ equals r(p) and that this also implies that the Iwasawa invariant ν equals r(p).