Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T13:48:16.678Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  01 June 2016

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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Copyright
© The Author(s) 2016 

References

Aoki, M. and Fukuda, T., ‘An algorithm for computing p-class groups of abelian number fields’, Algorithmic number theory , Lecture Notes in Computer Science 4076 (Springer, Berlin, 2006) 5671.Google Scholar
Bernstein, D., ‘Fast multiplication and its applications’, Algorithmic number theory: lattices, number fields, curves and cryptography , Mathematical Sciences Research Institute Publications 44 (Cambridge University Press, Cambridge, 2008) 325384.Google Scholar
Biasse, J.-F. and Fieker, C., ‘New techniques for computing the ideal class group and a system of fundamental units in number theory’, ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium (Mathematical Sciences Publishers, Berkeley, CA, 2013).Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) 235265.Google Scholar
Cohen, H., A course in computational algebraic number theory , Graduate Texts in Mathematics 138 (Springer, Berlin, 1993).Google Scholar
Cohen, H., Advanced topics in computational number theory , Graduate Texts in Mathematics 193 (Springer, New York, 2000).Google Scholar
Cohen, H., Number theory. Vol. II. Analytic and modern tools , Graduate Texts in Mathematics 240 (Springer, New York, 2007).Google Scholar
Gras, G. and Gras, M.-N., ‘Calcul du nombre de classes et des unités des extensions abéliennes réelles de Q’, Bull. Sci. Math. (2) 101 (1977) no. 2, 97129.Google Scholar
Hakkarainen, T., ‘On the computation of class numbers of real abelian fields’, Math. Comp. 78 (2009) no. 265, 555573.Google Scholar
Iwasawa, K., Lectures on p-adic L-functions , Annals of Mathematics Studies 74 (Princeton University Press/University of Tokyo Press, Princeton, NJ/Tokyo, 1972).CrossRefGoogle Scholar
Mäki, S., ‘The conductor density of abelian number fields’, J. Lond. Math. Soc. (2) 47 (1993) no. 1, 1830.Google Scholar
Neukirch, J., Algebraic number theory , Grundlehren der Mathematischen Wissenschaften 322 (Springer, Berlin, 1999); translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G Harder.Google Scholar
Schoof, R., ‘Class numbers of real cyclotomic fields of prime conductor’, Math. Comp. 72 (2003) no. 242, 913937.CrossRefGoogle Scholar
Washington, L. C., Introduction to cyclotomic fields , 2nd edn. Graduate Texts in Mathematics 83 (Springer, New York, 1997).CrossRefGoogle Scholar
Wright, D. J., ‘Distribution of discriminants of abelian extensions’, Proc. Lond. Math. Soc. (3) 58 (1989) no. 1, 1750.Google Scholar