Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Malle's conjecture for fair counting functions

Peter Koymans, Carlo Pagano

TL;DR

The paper shows that naively extending Malle's conjecture to fair counting functions (counting by the product of ramified primes) fails in general by constructing an infinite family of nilpotent groups of class 2 for which the expected logarithmic exponent undercounts the true growth. Central to the argument is a lifting phenomenon: after fixing a suitable cyclotomic subextension, every abelian quotient lift lifts to a full G-extension, enabling an abundance of G-extensions and yielding explicit asymptotics that violate the naive exponent. The authors derive a detailed counting mechanism using a Koymans–Pagano parametrization and compute the naive Malle constant, subsequently showing the exponent mismatch for the constructed groups. To remedy this, they propose a modified Malle conjecture based on the set $\mathrm{Epi}_{(H,\phi)}(G_{\mathbb{Q}}, G)$ with a fixed cyclotomic subextension, and they express the leading constant as an Euler product, introducing $b_{(H,\phi)}(G)$ to capture the refined logarithmic growth. This framework preserves desirable properties (Euler-product leading constants, removal of the subfield problem) in the fair-counting setting and generalizes existing conjectures to accommodate cyclotomic entanglement and local conditions.

Abstract

We show that the naive adaptation of Malle's conjecture to fair counting functions is not true in general.

Malle's conjecture for fair counting functions

TL;DR

The paper shows that naively extending Malle's conjecture to fair counting functions (counting by the product of ramified primes) fails in general by constructing an infinite family of nilpotent groups of class 2 for which the expected logarithmic exponent undercounts the true growth. Central to the argument is a lifting phenomenon: after fixing a suitable cyclotomic subextension, every abelian quotient lift lifts to a full G-extension, enabling an abundance of G-extensions and yielding explicit asymptotics that violate the naive exponent. The authors derive a detailed counting mechanism using a Koymans–Pagano parametrization and compute the naive Malle constant, subsequently showing the exponent mismatch for the constructed groups. To remedy this, they propose a modified Malle conjecture based on the set with a fixed cyclotomic subextension, and they express the leading constant as an Euler product, introducing to capture the refined logarithmic growth. This framework preserves desirable properties (Euler-product leading constants, removal of the subfield problem) in the fair-counting setting and generalizes existing conjectures to accommodate cyclotomic entanglement and local conditions.

Abstract

We show that the naive adaptation of Malle's conjecture to fair counting functions is not true in general.
Paper Structure (10 sections, 9 theorems, 102 equations)

This paper contains 10 sections, 9 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Malle's conjecture
  3. Fair counting functions
  4. Results and conjectures
  5. Comparison with previous results and conjectures
  6. Overview of the paper
  7. The construction
  8. Finding many lifts
  9. Counting conjugacy classes
  10. A modified Malle conjecture

Key Result

Theorem 1.3

There exists an infinite family of nilpotent groups $G$ of nilpotency class $2$ such that Conjecture cMalle fails for the pair $(G, \mathbb{Q})$. More precisely, the constant $b(G, \mathbb{Q})$ in equation (eMalleb) is too small.

Theorems & Definitions (24)

  • Conjecture 1.1: Malle2
  • Conjecture 1.2: Folklore adaptation of Malle's conjecture
  • Theorem 1.3
  • Conjecture 1.4: Conjecture \ref{['conj1']}
  • Conjecture 1.5: Conjecture \ref{['conj2']}
  • Theorem 1.6: Theorem \ref{['th: counterexample']}
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Lemma 3.2
  • ...and 14 more