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Statistics of bad parts of class groups

Peter Koymans, Yuan Liu

TL;DR

This work develops a statistical genus theory for the $p$-parts of class groups in abelian $p$-extensions. By translating class-group data into generalized Selmer groups with stringent local conditions, the authors prove that the dual Selmer group vanishes for 100% of extensions when ordered by the product of ramified primes, yielding an exact algebraic formula for the $I$-rank of $e\mathrm{Cl}(K)$ in terms of a combinatorial set of special primes and a cohomology-dependent constant. The main technical engine combines local condition analysis, Greenberg–Wiles, and a detailed character-sum and large-sieve framework, together with a thorough combinatorial classification of unlinked index sets. They also provide explicit corollaries for cyclic and elementary abelian $p$-groups, highlighting sharp connections between ramification data and the growth of $p$-parts of class groups. Overall, the results extend genus theory to a broad, probabilistic setting and furnish deterministic leading terms in a statistical context, with potential applications to broader arithmetic-statistics problems.

Abstract

Let $p$ be an odd prime. We give a formula for the bad part of $p$-class groups that is valid for $100\%$ of the abelian $p$-extensions when ordered by product of ramified primes.

Statistics of bad parts of class groups

TL;DR

This work develops a statistical genus theory for the -parts of class groups in abelian -extensions. By translating class-group data into generalized Selmer groups with stringent local conditions, the authors prove that the dual Selmer group vanishes for 100% of extensions when ordered by the product of ramified primes, yielding an exact algebraic formula for the -rank of in terms of a combinatorial set of special primes and a cohomology-dependent constant. The main technical engine combines local condition analysis, Greenberg–Wiles, and a detailed character-sum and large-sieve framework, together with a thorough combinatorial classification of unlinked index sets. They also provide explicit corollaries for cyclic and elementary abelian -groups, highlighting sharp connections between ramification data and the growth of -parts of class groups. Overall, the results extend genus theory to a broad, probabilistic setting and furnish deterministic leading terms in a statistical context, with potential applications to broader arithmetic-statistics problems.

Abstract

Let be an odd prime. We give a formula for the bad part of -class groups that is valid for of the abelian -extensions when ordered by product of ramified primes.
Paper Structure (33 sections, 27 theorems, 202 equations)

This paper contains 33 sections, 27 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Related results
  3. Methods
  4. Notation
  5. Local conditions
  6. Size of local conditions
  7. An explicit formula for the cardinality of local conditions
  8. Bounds for the local conditions of the dual Selmer group
  9. Vanishing of the dual Selmer group
  10. Expressing class groups as Selmer groups
  11. Algebraic formula for class groups
  12. Statistical formula for N
  13. Computing Ntypical(A, M) in special cases
  14. Character sums
  15. Abstract setup
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $p$ be an odd prime, $A$ a finite abelian $p$-group, and $e$ a nontrivial primitive idempotent of $\mathbb{Q}_p[A]$. For a proper ideal $I$ of $e\mathbb{Z}_p[A]$ containing $I_e$ and an $A$-extension $K/\mathbb{Q}$ with a chosen isomorphism $\iota: \mathrm{Gal}(K/\mathbb{Q}) \to A$, define a set where $D_v$ and $I_v$ are respectively the decomposition subgroup and inertia subgroup of $K/\mathb

Theorems & Definitions (65)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6
  • ...and 55 more