A Secondary Term for $D_4$ Quartic Fields Ordered By Conductor

TL;DR

This paper refines the counting of $D_4$ quartic fields ordered by conductor by identifying a secondary term and improving the error estimate, advancing understanding of their distribution.

Contribution

It introduces a secondary term and power-saving error term in the asymptotic count of $D_4$ quartic fields ordered by conductor, refining previous results.

Findings

Identified a secondary term in the asymptotic count.

Achieved a power-saving error estimate.

Enhanced understanding of $D_4$ quartic fields distribution.

Abstract

To any quartic $D_4$ extension of $\mathbb{Q}$, one can associate the Artin conductor of a 2-dimensional irreducible representation of the group. Alt\u{u}g, Shankar, Varma, and Wilson determined the asymptotic number of such fields when ordered by conductor. We refine this, realizing a secondary term and power saving error term.