Arakelov class groups of random number fields

Mathematische Annalen, vol. 390, pp. 4405–4428

Abstract

Abstract The main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen–Lenstra–Martinet heuristic on ideal class groups. To that end, we show that Chinburg’s $$\Omega (3)$$ Ω ( 3 ) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups. As a consequence, we construct a new infinite series of counterexamples to the Cohen–Lenstra–Martinet heuristic, which have the novel feature that their Galois groups are non-abelian.

Authors 3

1. Alex Bartel corresponding

   University of Glasgow

   Affiliation as printed

   School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8SQ, UK

2. Henri Johnston

   University of Exeter

   Affiliation as printed

   Department of Mathematics and Statistics, University of Exeter, Exeter, EX4 4QF, UK

3. Hendrik W. Lenstra Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA, Leiden, The Netherlands