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Arakelov class groups of random number fields

Alex Bartel, Henri Johnston, Hendrik W. Lenstra

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) 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.

Arakelov class groups of random number fields

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) 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.

Paper Structure

This paper contains 11 sections, 26 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pontryagin duality
  4. The (oriented) Arakelov class group
  5. Modules and Grothendieck groups
  6. Cohen--Lenstra--Martinet heuristic
  7. Grothendieck groups of orders
  8. Oriented Arakelov class groups as Galois modules
  9. The relation to Chinburg's $\Omega(3)$ conjecture
  10. Some known cases of the conjectures
  11. Counterexamples to the Cohen--Lenstra--Martinet heuristic

Key Result

Theorem 1.1

With the notation just introduced, suppose that Chinburg's $\Omega(3)$ conjecture, Conjecture conj:omega-3, holds for $F/K$. Suppose, moreover, that for every prime number $p$ not dividing $2\cdot\#G$, each primitive $p$-th root of unity in $F$ is in $K$. Then the equality holds in $\mathop{\mathrm{G}}\nolimits_{0}(\Lambda)_{\mathop{\mathrm{tors}}\nolimits}$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 3.1
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • proof
  • ...and 52 more