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Counterexamples for Türkelli's Modification on Malle's Conjecture

Jiuya Wang

TL;DR

The paper presents counterexamples to Türkelli's modification of Malle's conjecture, showing that the proposed $b$-constant $b_T(G,k)$ can differ from the true $b(G,k)$ and from Malle's original prediction, with explicit instances where function-field and number-field constants diverge. It analyzes the embedding problem of cyclotomic extensions, introduces a refined, invariant-sensitive framework that partitions G-extensions by cyclotomic data, and defines a lifting-based counting function $N_{Q,\pi,\phi}(G^{inv},X)$ to capture these subtleties. The authors compute $b$-constants for specific wreath-product groups over both number fields and function fields, demonstrating infinite families where $b_T$ and $b_M$ bound the true constant in different ways and where $b$-constants diverge across base fields. They propose a refined Malle conjecture that aggregates over finitely many embedding data pairs $(\pi,\phi)$ and accommodates function-field behavior, supported by function-field results and recent advances on embedding problems. Overall, the work indicates that a universal, single-constant formulation of Malle’s conjecture is unlikely and motivates a composition-by-embedding approach to capture cyclotomic interactions and constants more accurately, potentially guiding future conjectures and proofs in both arithmetic and function-field contexts.

Abstract

We give counterexamples for the modification on Malle's Conjecture given by Türkelli. Türkelli's modification on Malle's conjecture is inspired by an analogue of Malle's conjecture over a function field. As a consequence, our counterexamples demonstrate that the $b$ constant can differ between function fields and number fields. We also show that Klüners' counterexamples give counterexamples for a natural extension of Malle's conjecture to counting number fields by product of ramified primes. We then propose a refined version of Malle's conjecture which implies a new conjectural value for the constant $b$ for number fields.

Counterexamples for Türkelli's Modification on Malle's Conjecture

TL;DR

The paper presents counterexamples to Türkelli's modification of Malle's conjecture, showing that the proposed -constant can differ from the true and from Malle's original prediction, with explicit instances where function-field and number-field constants diverge. It analyzes the embedding problem of cyclotomic extensions, introduces a refined, invariant-sensitive framework that partitions G-extensions by cyclotomic data, and defines a lifting-based counting function to capture these subtleties. The authors compute -constants for specific wreath-product groups over both number fields and function fields, demonstrating infinite families where and bound the true constant in different ways and where -constants diverge across base fields. They propose a refined Malle conjecture that aggregates over finitely many embedding data pairs and accommodates function-field behavior, supported by function-field results and recent advances on embedding problems. Overall, the work indicates that a universal, single-constant formulation of Malle’s conjecture is unlikely and motivates a composition-by-embedding approach to capture cyclotomic interactions and constants more accurately, potentially guiding future conjectures and proofs in both arithmetic and function-field contexts.

Abstract

We give counterexamples for the modification on Malle's Conjecture given by Türkelli. Türkelli's modification on Malle's conjecture is inspired by an analogue of Malle's conjecture over a function field. As a consequence, our counterexamples demonstrate that the constant can differ between function fields and number fields. We also show that Klüners' counterexamples give counterexamples for a natural extension of Malle's conjecture to counting number fields by product of ramified primes. We then propose a refined version of Malle's conjecture which implies a new conjectural value for the constant for number fields.

Paper Structure

This paper contains 16 sections, 21 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Malle's Conjecture
  3. Türkelli's Modification: inspiration and comparison with function fields
  4. General Invariants
  5. A New Proposal
  6. Description of $b(G,k)$
  7. Malle's constant $b_M(G,k)$
  8. Türkelli's constant $b_T(G,k)$
  9. Computation of $b(G,Q)$
  10. Group Constant Computation
  11. Field Counting: Number Fields
  12. Field Counting: Global Function Fields
  13. Embedding Cyclotomic Extensions
  14. Function Fields
  15. Number Fields
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $\ell$ be an odd prime number and $d = \prod_i p_i^{r_i} \neq 2$ where $p_i$ are all prime numbers. Let $G = C_{\ell} \wr C_{d}\subset S_{\ell d}$ with $(|G|, \operatorname{char}(Q)) = 1$, $\mathop{\mathrm{Gal}}\nolimits({\mathbb{F}}_q(t)(\mu_{\ell})/{\mathbb{F}}_q(t)) \simeq \mathop{\mathrm{Gal As a consequence, there exists $G\subset S_n$ such that $\mathop{\mathrm{Gal}}\nolimits({\mathbb{F}

Theorems & Definitions (65)

  • Conjecture 1: Malle's Conjecture over Number Fields, Mal02Mal04
  • Conjecture 2: Malle's Conjecture over Function Field
  • Conjecture 3: Türkelli's Modification turkelli2015connected
  • Example 1.1
  • Theorem 1.3
  • Conjecture 4
  • Conjecture 5: Generalized Malle's Conjecture
  • Definition 1.4: Concentrated Group, ALOWW
  • Definition 1.5: Big Fiber
  • Theorem 1.6
  • ...and 55 more