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A family of pairs of imaginary cyclic fields of degree \((p-1)/2\) with both class numbers divisible by p

  • Miho AokiEmail author
  • Yasuhiro Kishi
Article
  • 61 Downloads

Abstract

Let p be a prime number with \(p\equiv 5\ (\mathrm{mod}\ {8})\). We construct a new infinite family of pairs of imaginary cyclic fields of degree \((p-1)/2\) with both class numbers divisible by p. Let \(k_0\) be the unique subfield of \(\mathbb {Q}(\zeta _p)\) of degree \((p-1)/4\) and \(u_p=(t+b\sqrt{p})/2\,(>1)\) be the fundamental unit of \(k:=\mathbb {Q}(\sqrt{p})\). We put \(D_{m,n}:={\mathcal {L}}_m(2{\mathcal {F}}_m-{\mathcal {F}}_n{\mathcal {L}}_m)b\) for integers m and n, where \(\{ {\mathcal {F}}_n \}\) and \(\{ {\mathcal {L}}_n \}\) are linear recurrence sequences of degree two associated to the characteristic polynomial \(P(X)=X^2-tX-1\). We assume that there exists a pair \((m_0,n_0)\) of integers satisfying certain congruence relations. Then we show that there exists a positive integer \(N_q\) which satisfies the both class numbers of \(k_0(\sqrt{D_{m,n}})\) and \(k_0(\sqrt{pD_{m,n}})\) are divisible by p for any pairs (mn) with \(m\equiv m_0 \ (\mathrm{mod}\ {N_q}), \ n\equiv n_0 \ (\mathrm{mod}\ {N_q})\) and \(n>3\). Furthermore, we show that if we assume that ERH holds, then there exists the pair \((m_0,n_0)\).

Keywords

Class numbers Abelian number fields Fundamental units Gauss sums Jacobi sums Linear recurrence sequences 

Mathematics Subject Classification

11R11 11R16 11R29 

Notes

Acknowledgements

The authors would like to thank Toru Komatsu and the referee for useful advices. They would also like to thank Takuya Yamauchi for his polite suggestions on the proof of Lemma 15.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Interdisciplinary Faculty of Science and EngineeringShimane UniversityMatsueJapan
  2. 2.Department of Mathematics, Faculty of EducationAichi University of EducationKariyaJapan

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