[Paper Review] The number of quartic $D_4$-fields ordered by conductor

Salim Ali Altuğ, Arul Shankar+2 more|arXiv (Cornell University)|Apr 6, 2017

Algebraic Geometry and Number Theory14 references6 citations

TL;DR

This paper determines the asymptotic number of quartic $D_4$-fields ordered by conductor, establishing an explicit mass formula for the leading term. By combining arithmetic invariant theory with L-function methods and exploiting the outer automorphism of $D_4$, the authors derive precise asymptotics that verify Kedlaya and Wood's heuristics and extend to counting order-4 elements in class and narrow class groups of quadratic fields ordered by discriminant.

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.

Motivation & Objective

• To determine the asymptotic number of quartic $D_4$-fields with bounded conductor, refining previous results that ordered by discriminant.
• To verify the heuristics of Kedlaya and Wood for $D_4$-fields by deriving an explicit mass formula for the leading term.
• To extend the counting method to include local splitting conditions at finitely or infinitely many primes, enabling applications to class group statistics.
• To compute the asymptotic number of order-4 elements in class groups and narrow class groups of quadratic fields, ordered by discriminant.
• To resolve the discrepancy between discriminant-based and conductor-based counting by showing that the conductor-based constant arises from a consistent Euler product mass formula.

Proposed method

• The authors use the Artin conductor of the 2-dimensional irreducible Galois representation with image $D_4$ as the primary invariant for ordering fields.
• They combine arithmetic invariant theory with geometry-of-numbers techniques and L-function methods, enhanced by the outer automorphism of $D_4$ to overcome difficulties in counting fields with large quadratic subfield discriminants.
• A key innovation is the use of local specifications $\Sigma^{(ullet)}$ at finite and infinite primes to impose splitting conditions, enabling control over unramified extensions and class group elements.
• The method involves constructing a stable and acceptable collection of local conditions $( ho_p, ho_p')$ at each prime, ensuring compatibility with the Galois group structure of $D_4$.
• The asymptotic counts are derived via density computations and the application of Proposition 6.7, linking the number of $D_4$-fields to class group data.
• The leading term is expressed as an Euler product over primes: $\prod_p \left(1 - \frac{1}{p^2} - \frac{2}{p^3} + \frac{2}{p^4}\right)$, which is shown to be a mass formula.

Experimental results

Research questions

• RQ1 What is the asymptotic number of $D_4$-quartic fields with conductor bounded by $X$, for each signature (0,2), (1,1), and (2,0)?
• RQ2 Can the leading constant in the asymptotic count of $D_4$-fields ordered by conductor be expressed as a mass formula, verifying Kedlaya and Wood's heuristics?
• RQ3 How does the conductor-based counting of $D_4$-fields relate to the distribution of order-4 elements in class groups and narrow class groups of quadratic fields?
• RQ4 Can local splitting conditions at finitely or infinitely many primes be incorporated into the counting, and what are the resulting asymptotics?
• RQ5 Why does the discriminant-based constant for $D_4$-fields differ from the conductor-based one, and can the latter be expressed as a consistent Euler product?

Key findings

• The number of totally real $D_4$-fields with conductor $\leq X$ is asymptotically $\frac{1}{4} \cdot \prod_p \left(1 - \frac{1}{p^2} - \frac{2}{p^3} + \frac{2}{p^4}\right) \cdot X \log X + O(X \log \log X)$.
• The number of complex $D_4$-fields with conductor $\leq X$ is asymptotically $\frac{3}{8} \cdot \prod_p \left(1 - \frac{1}{p^2} - \frac{2}{p^3} + \frac{2}{p^4}\right) \cdot X \log X + O(X \log \log X)$.
• The number of totally complex $D_4$-fields with conductor $\leq X$ is asymptotically $\frac{1}{8} \cdot \prod_p \left(1 - \frac{1}{p^2} - \frac{2}{p^3} + \frac{2}{p^4}\right) \cdot X \log X + O(X \log \log X)$.
• The leading constant in the asymptotic is shown to be a mass formula, resolving the discrepancy between discriminant-based and conductor-based counts for $D_4$-fields.
• The method allows counting $D_4$-fields with specified local splitting behavior, enabling the asymptotic count of order-4 elements in class groups of quadratic fields ordered by discriminant.
• The asymptotic for order-4 elements in narrow class groups of real quadratic fields is derived via the $\Sigma^{(\text{c})}$-conditions, confirming consistency with class field theory and genus theory.