Kervaire--Murthy conjecture, Ullom's inequality and Iwaswa numbers

Details Kervaire--Murthy conjecture, Ullom's inequality and Iwaswa numbers Kervaire--Murthy conjecture, Ullom's inequality and Iwaswa numbers

2010-03-09

arxiv.org/abs/1003.1871v3

Mathematics

Number Theory

12 pages

Scientific paper

We study the cyclotomic field of $p^n$ roots of unity and the Sylow p-component of its class group. Here $p$ is a semi-regular prime. We prove that for $n\geq 2$ the number of generators of this group is equal to the corresponding Iwasawa number $\lambda$. Simultaneously we prove that the Iwasawa numbers $\lambda_k$ for the field of rational numbers are less than $p$. Under the assumption that this inequality was true, recently Tauno Mets\"ankyl\"a (J. Number Theory 130(2010), 727--737) obtained results about behavior of the $p$-adic $L$-function $L_p (s,\omega^{k})$ and certain congruences of a new type for (generalized) Bernoulli numbers. We discuss these results in Appendix.

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