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A note on the Demjanenko matrices related to the cyclotomic \({\mathbb{Z}}_p\)-extension. (English) Zbl 0965.11044

For any odd prime number \(p\), the Maillet determinant \(D(p)\) is related to the relative class number of the cyclotomic field \(\mathbb{Q}(\zeta_p)\) by the formula of Carlitz and Olson: \[ D(p)= (-p)^{(p-3)/2} h^- (\mathbb{Q}(\zeta_p)). \] This has been generalized to imaginary abelian fields by Tateyama, Girstmair, etc. …
The determinant of the Demjanenko matrix \(M(p)\) is analogously related to \(h^-(\mathbb{Q} (\zeta_p))\). This has been generalized to imaginary abelian fields by Tsumura, Hirabayashi, etc. … In this note, the author introduces the cyclotomic \(\mathbb{Z}_p\)-extension \(K_\infty= \bigcup_m K_m\) of an imaginary abelian field with conductor \(dp\), \(p\nmid d\) and generalized Demjanenko matrices \(\Delta(K,\ell,m)\), \((\ell,dp)=1\); the main result is a formula relating \(\det \Delta(K,\ell,m)\) to \(h^-(K_m)\).

MSC:

11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
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References:

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