Secondary terms in the counting functions of quartic fields II

TL;DR

This paper refines the counting of $S_4$-quartic fields with bounded discriminant, providing precise asymptotics with error terms, and analyzes related zeta functions to understand their pole structure.

Contribution

It offers new asymptotic formulas for counting $S_4$-quartic fields with local conditions and establishes properties of associated zeta functions, advancing understanding of quartic field distributions.

Findings

Precise asymptotic counts with power-saving error terms.

Determination of the pole structure of related zeta functions.

Extension of counting results to quartic rings with local conditions.

Abstract

We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for quartic rings (weighted by the number of cubic resolvents), deducing as a consequence that the Shintani zeta functions associated to the prehomogeneous vector space $\mathbb{C}^2\otimes\mathrm{Sym}^2(\mathbb{C}^3)$ have at most a simple pole at $s=5/6$.