On the Spiegelungssatz for the 4-rank

Fouvry, Étienne; Klüners, Jürgen

doi:10.2140/ant.2010.4.493

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On the Spiegelungssatz for the 4-rank

Étienne Fouvry and Jürgen Klüners

Vol. 4 (2010), No. 5, 493–508

DOI: 10.2140/ant.2010.4.493

Abstract

Let $d$ be a nonsquare positive integer. We give the value of the natural probability that the narrow ideal class groups of the quadratic fields $ℚ (\sqrt{d})$ and $ℚ (\sqrt{- d})$ have the same $4$ -ranks.

Keywords

quadratic fields, class groups, reflection principle

Mathematical Subject Classification 2000

Primary: 11R29

Secondary: 11R11

Milestones

Received: 2 October 2008

Revised: 30 November 2009

Accepted: 19 January 2010

Published: 10 July 2010

Authors

Étienne Fouvry
Université de Paris-Sud Laboratoire de Mathématiques 91405 Orsay Cedex France	CNRS 91405 Orsay Cedex France

Jürgen Klüners
Universität Paderborn Institut für Mathematik Warburger Straße 100 33098 Paderborn Germany

Vol. 4, No. 5, 2010