Malle's conjecture for nonic Heisenberg extensions

TL;DR

This paper proves Malle's conjecture for nonic Heisenberg extensions over the rationals, showing the count of such extensions with bounded discriminant relates to character sums and analyzing their oscillations.

Contribution

It establishes Malle's conjecture for nonic Heisenberg extensions over ield, using character sums and oscillation analysis to determine the asymptotic count.

Findings

Number of nonic Heisenberg extensions with discriminant ounded by X is given by a character sum.

Main term of the count is extracted by analyzing character oscillations.

The result confirms Malle's conjecture in this specific nonic Heisenberg setting.

Abstract

We prove Malle's conjecture for nonic Heisenberg extensions over $\mathbb{Q}$. Our main algebraic result shows that the number of nonic Heisenberg extensions over $\mathbb{Q}$ with discriminant bounded by $X$ is given by a character sum. We then extract the main term from this sum by exploiting oscillation of characters.