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 \(h^-(K_m)\).


11R23 Iwasawa theory
11R20 Other abelian and metabelian extensions
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[1] Hazama, F.: Demjanenko matrix, class number, and Hodge group. J. Number Thoery, 34 , 174-177 (1990). · Zbl 0697.12003 · doi:10.1016/0022-314X(90)90147-J
[2] Hirabayashi, M.: A generalization of Maillet and Demyanenko determinants. Acta Arith., 83 , 391-397 (1998). · Zbl 0895.11045
[3] Girstmair, K.: The relative class numbers of imaginary cyclotomic fields of degrees 4, 6, 8, and 10. Math. Comp., 61 , 881-887 (1993). · Zbl 0787.11046 · doi:10.2307/2153259
[4] Sands, J. W., and Schwarz, W.: A Demjanenko matrix for abelian fields of prime power conductor. J. Number Theory, 52 , 85-97 (1995). · Zbl 0829.11054 · doi:10.1006/jnth.1995.1057
[5] Tateyama, K.: Maillet’s determinant. Sci. Papers College Gen. Ed. Univ. Tokyo, 32 , 97-100 (1982). · Zbl 0508.12006
[6] Tsumura, H.: On Demjanenko’s matrix and Maillet’s determinant for imaginary abelian number fields. J. Number Theory, 60 , 70-80 (1996). · Zbl 0866.11063 · doi:10.1006/jnth.1996.0113
[7] Washington, L. C.: Introduction to Cyclotomic Fields. Springer, New-York (1982). · Zbl 0484.12001
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