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On ${\it\lambda}$-invariants attached to cyclic cubic number fields

Published online by Cambridge University Press:  01 December 2015

Daniel Delbourgo
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand email
Qin Chao
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand email


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We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$, where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$-invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$. If the ${\it\lambda}$-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$-adic $L$-function and deduce ${\rm\Lambda}$-monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.

Supplementary materials are available with this article.

Research Article
© The Author(s) 2015 


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