2002/07/30 by Ola Helenius, Helenius, Ola, Alexander Stolin +1
Mathematics · Social Sciences · #11R21 #11R65 #19A31 #African history and culture studies #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Geopolitical and Social Dynamics #K-Theory and Homology (math.KT) #Number Theory (math.NT) #math.KT #math.NT #msc:11R21 #msc:11R65 #msc:19A31
paper · pdf · doi:10.48550/arxiv.math/0207286
34 pages, new version with some typos corrected
openalex publication_date 2002/07/30 · arxiv created 2002/09/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→ \pic (ℤ Cpn)→ \cl ℚ (ζn-1)× \pic (ℤ Cpn-1)→ 0 where ζn-1 is a primitive pn-th root of unity. The group Vn that injects into \pic ℤ Cpn≅K0ℤ 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 \Z 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, \charVn+≅ \cl(p) ℚ (ζn-1), the p-part of the class group. In the present paper we calculate Vn+ and prove that \char Vn+≅ \cl(p) ℚ(ζn-1) for all semi-regular primes which also gives us the structure of \cl(p) ℚ(\zn-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).