The number of quartic $D_4$-fields ordered by conductor

TL;DR

This paper determines the asymptotic count of quartic $D_4$-fields ordered by conductor, providing explicit formulas, verifying heuristics, and allowing local condition impositions, thus advancing understanding of these number fields.

Contribution

It introduces a novel approach combining algebraic structure and existing methods to count $D_4$-quartic fields by conductor, overcoming previous counting difficulties.

Findings

Asymptotic formula for counting $D_4$-fields ordered by conductor

Explicit mass formula for the leading term

Ability to impose local splitting conditions at finitely and infinitely many primes

Abstract

We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image $D_4$. We determine the asymptotic number of such quartic $D_4$-fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order 4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of $D_4$-fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of $D_4$ combined with both approaches mentioned above to obtain exact asymptotics.