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Class number calculation using Siegel functions

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  01 August 2014

T. Fukuda  and
K. Komatsu
T. Fukuda
Affiliation:
Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan email fukuda.takashi@nihon-u.ac.jp
K. Komatsu
Affiliation:
Department of Mathematical Science, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan email kkomatsu@waseda.jp

Abstract

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We propose a fast method of calculating the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-part of the class numbers in certain non-cyclotomic $\mathbb{Z}_p$-extensions of an imaginary quadratic field using elliptic units constructed by Siegel functions. We carried out practical calculations for $p=3$ and determined $\lambda $-invariants of such $\mathbb{Z}_3$-extensions which were not known in our previous paper.

MSC classification

Secondary: 11R29: Class numbers, class groups, discriminants 11R23: Iwasawa theory

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