Counterexamples for T\"urkelli's Modification on Malle's Conjecture

TL;DR

This paper presents counterexamples to T"urkelli's modification of Malle's conjecture, showing differences between function fields and number fields, and proposes a refined conjecture with a new constant.

Contribution

It provides the first known counterexamples to T"urkelli's modification and introduces a refined version of Malle's conjecture with a new constant value.

Findings

Counterexamples show $b$ constant differs between function fields and number fields.

Kl"uners' counterexamples extend to a natural number field conjecture.

A refined conjecture with a new $b$ constant is proposed.

Abstract

We give counterexamples for the modification on Malle's Conjecture given by T\"urkelli. T\"urkelli'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 Kl\"uners' 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.