Triviality of Iwasawa module associated to some abelian fields of prime conductors

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Abstract

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.

Keywords

Iwasawa module Class number Cyclotomic field 

Mathematics Subject Classification

Primary 11R23 Secondary 11R18 

Notes

Acknowledgements

The author thanks the referee for several valuable comments which improved the presentation of the whole paper and for suggesting him the simple proofs of Lemma 1(II) and Lemma 5 and informing him of the paper of Hardy and Littlewood [11]

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Faculty of ScienceIbaraki UniversityMitoJapan

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