Fine Structure of Class Groups \cl^(p)\Q(\z_n) and the Kervaire--Murthy Conjectures II

2002/09/06 by Ola Helenius, Helenius, Ola, Alexander Stolin +1
Arts and Humanities · Mathematics · Social Sciences · #11R21 #11R65 #19A31 #African history and culture studies #FOS: Mathematics #Historical Linguistics and Language Studies #Historical Studies and Socio-cultural Analysis #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/0209066

7 pages, Continuation of NT/0207286

arxiv created 2002/09/06 · openalex publication_date 2002/09/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There is an Mayer-Vietoris exact sequence involving the Picard group of the integer group ring \Z C_pⁿ where C_pⁿ is the cyclic group of order pⁿ and ζ_n-1 is a primitive pⁿ-th root of unity. The "unknown" part of the sequence is a group. V_n. V_n splits as V_n≅ V_n⁺⊕ V_n^- and V_n^- is explicitly known. V_n⁺ is a quotient of an in some sense simpler group V_n. In 1977 Kervaire and Murthy conjectured that for semi-regular primes p, V_n⁺ ≅ V_n⁺ ≅ \cl^(p)(\Q (ζ_n-1))≅ (ℤ/pⁿ ℤ)^r(p), 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/pⁿ ℤ)^r(p)⊕ (ℤ/p^n-1 ℤ)^λ-r(p). In the previous paper we proved that for all semi-regular primes, V_n⁺≅ \cl^(p)(\Q (ζ_n-1)) and that these groups are isomorphic to (ℤ/pⁿ ℤ)^r₀⊕ (ℤ/p^n-1 ℤ)^r₁-r₀ ⊕ \hdots ⊕ (ℤ/p ℤ)^r_n-1-r_n-2 for a certain sequence \r_k\ (where r₀=r(p)). Under Ulloms extra condition it was proved that V_n⁺ ≅ V_n⁺ ≅ \cl^(p)(\Q(\z_n-1)) ≅ (ℤ/pⁿ ℤ)^r(p)⊕ (ℤ/p^n-1ℤ)^λ-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.

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