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Application of the Strong Artin Conjecture to the Class Number Problem

Published online by Cambridge University Press:  20 November 2018

Peter J. Cho  and
Henry H. Kim
Peter J. Cho
Affiliation:
Department of Mathematics, University of Toronto, TorontoON M5S 2E4, e-mail: jcho@math.toronto.edu
Henry H. Kim
Affiliation:
Department of Mathematics, University of Toronto, TorontoON M5S 2E4 andKorea Institute for Advanced Study, Seoul, Korea, e-mail: henrykim@math.toronto.edu
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Abstract

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We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group ${{A}_{4}},\,{{S}_{4}}$, and ${{S}_{5}}$. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying the zero density result of Kowalski–Michel, we choose subfamilies of $L$-functions that are zero-free close to 1. For these subfamilies, the $L$-functions have the extremal value at $s\,=\,1$, and by the class number formula, we obtain the extreme class numbers.

Keywords

11R2911M41class numberstrong Artin conjecture
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 6 , 01 December 2013 , pp. 1201 - 1216
Copyright
Copyright © Canadian Mathematical Society 2013

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