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An application of the $p$-adic analytic class number formula

Part of: Computational number theory Number theory Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  01 June 2016

Claus Fieker  and
Yinan Zhang
Claus Fieker
Affiliation:
Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany email fieker@mathematik.uni-kl.de
Yinan Zhang
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Australia email y.zhang@sydney.edu.au

Abstract

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We propose an algorithm to verify the $p$-part of the class number for a number field $K$, provided $K$ is totally real and an abelian extension of the rational field $\mathbb{Q}$, and $p$ is any prime. On fields of degree 4 or higher, this algorithm has been shown heuristically to be faster than classical algorithms that compute the entire class number, with improvement increasing with larger field degrees.

MSC classification

Primary: 11S40: Zeta functions and $L$-functions
Secondary: 11-04: Explicit machine computation and programs (not the theory of computation or programming) 11Y35: Analytic computations 11Y70: Values of arithmetic functions; tables
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 217 - 228
Copyright
© The Author(s) 2016 

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