On Kervaire--Murthy conjecture, Bernoulli and Iwasawa numbers, and zeroes of p-adic L-function

2010/03/09 by Alexander Stolin, Stolin, Alexander
Mathematics · #11R23 #11R29. Secondary 19A31 #FOS: Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT) #math.KT #math.NT #msc:11R23 #msc:11R29. #msc:19A31

paper · pdf · doi:10.48550/arxiv.1003.1871

27 pages

arxiv created 2021/09/10 · arxiv updated 2021/09/13

Abstract

The aim of the present paper is to establish relations between Iwasawa and Bernoulli numbers based on some results by M. Kervaire and M. P. Murthy about the structure of the K₀ groups of the integer group rings of cyclic groups of prime power order pⁿ . In particular, we will prove that λ_i≤ p-1 under assumption that the generalized Bernoulli number B_1,ω^-i is not divisible by p². Here ω is the Teichmüller character of ℤ/(p-1)ℤ. λ_i=1 if B_1,ω^-i is divisible by p². We will prove that S_n,i≅ ℤ/(p^n+k_i), where S_n is the Sylow p-subgroup of the class group of the field ℚ(ζ_n). Here, ζ_n is a primitive pⁿ⁺¹-root of unity, ε_i are idempotents in the group ring \mathbb Z_p[\rm Gal(ℚ (ζ₀) /ℚ)], S_n,i=ε_i (S_n), and k_i is the p-adic valuation of B_1,ω^-i. At the end we will prove that k_i ≤ 1 and also v_p (L_p (0, ω^j))≤ 1 for even j under certain conditions on zeroes of L_p (0, ω^j) . Throughout the paper we assume that p satisfies Vandiver's conjecture.

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