The local zeta function in enumerating quartic fields

TL;DR

This paper derives an exact formula for the Fourier transform of the local maximality condition in a specific prehomogeneous vector space, advancing the enumeration of quartic fields by solving the local case for primes greater than 3.

Contribution

It provides the first explicit formula for the local Fourier transform related to the maximality condition in quartic field enumeration.

Findings

Exact formula for the Fourier transform of the local maximality condition.

Solves the local 'quartic case' in enumerating quartic fields.

Enhances understanding of local conditions in number field enumeration.

Abstract

An exact formula is obtained for the Fourier transform of the local condition of maximality modulo primes $p>3$ in the prehomogeneous vector space $2 \otimes \mathrm{Sym}^2(\mathbb{Z}_p^3)$ parametrizing quartic fields, thus solving the local `quartic case' in enumerating quartic fields.