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ON THE $4$-RANK OF CLASS GROUPS OF DIRICHLET BIQUADRATIC FIELDS

Part of: Multiplicative number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  22 December 2020

Étienne Fouvry ,
Peter Koymans Open the ORCID record for Peter Koymans [Opens in a new window]  and
Carlo Pagano
Étienne Fouvry
Affiliation:
Université Paris–Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405Orsay, France ([email protected])
Peter Koymans
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111Bonn, Germany ([email protected])
Carlo Pagano
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111Bonn, Germany ([email protected])
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Abstract

We show that for $100\%$ of the odd, square free integers $n> 0$ , the $4$ -rank of $\text {Cl}(\mathbb{Q} (i, \sqrt {n}))$ is equal to $\omega _3(n) - 1$ , where $\omega _3$ is the number of prime divisors of n that are $3$ modulo $4$ .

Keywords

class groupsarithmetic statisticsbiquadratic fields

MSC classification

Primary: 11N45: Asymptotic results on counting functions for algebraic and topological structures
Secondary: 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1543 - 1570
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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