2,407 research outputs found

    Estimating class numbers over metabelian extensions

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    Let pp be an odd prime and L/KL/K a pp-adic Lie extension whose Galois group is of the form Zpd1Zp\mathbb{Z}_p^{d-1}\rtimes \mathbb{Z}_p. Under certain assumptions on the ramification of pp and the structure of an Iwasawa module associated to LL, we study the asymptotic behaviours of the size of the pp-primary part of the ideal class groups over certain finite subextensions inside L/KL/K. This generalizes the classical result of Iwasawa and Cuoco-Monsky in the abelian case and gives a more precise formula than a recent result of Perbet in the non-commutative case when d=2d=2.Comment: 16 page

    Plus and minus logarithms and Amice transform

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    We give a new description of Pollack's plus and minus pp-adic logarithms logp±\log_p^\pm in terms of distributions. In particular, if μ±\mu_\pm denote the pre-images of logp±\log_p^\pm under the Amice transform, we give explicit formulae for the values μ±(a+pnZp)\mu_\pm(a+p^n\mathbb{Z}_p) for all aZpa\in \mathbb{Z}_p and all integers n1n\ge1. Our formulae imply that the distribution μ\mu_- agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.Comment: 9 page