The number of $D_4$-fields ordered by conductor

Abstract

We consider families of quartic number fields 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 D_4 -quartic 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 fail in the case of D_4 -quartic 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.

Geometry-of-numbers methods in the cusp and applications to class groups

“…By our explicit characterization of the elements of N (see (10) and ( 11)), the row-i, column-j entry of u is a non-constant polynomial in the unipotent coordinates for every pair (i, j) with i > j + 1. Thus, there exists an open subset U 3 ⊂ N of full measure such that for any u ∈ U 3 and u 0 ∈ N ∩ SL ± n (Z), the row-i, column-j entry of u 0 u is not an integer multiple of 1 2 for every pair (i, j) with i > j, unless i + 1 = j = n 2 , in which case u ij = 0. In particular, if for u ∈ U 3 and u 0 ∈ N ∩ SL ± n (Z) we have u 0 u ∈ N ′ , then u 0 u must in fact lie in the interior of N ′ , and imitating the proof of uniqueness in Lemma 15 yields that u 0 must be unique.…”

“…In certain special cases (see [1,15]), this approach has been generalized using situation-specific techniques to study objects parametrized by reducible orbits. The purpose of this article is to develop a systematic method for determining asymptotics for the number of reducible integral orbits of coregular representations.…”

“…A monogenized order comprises the data (O, α) of a monogenic order O = Z[α] in a number field together with a choice of monogenizer α ∈ O. 1 Two monogenized orders (O, α) and (O ′ , α ′ ) are declared isomorphic if there exists a ring isomorphism from O to O ′ taking α to ±α ′ + m for some m ∈ Z. If (O, α) is a monogenized order, its isomorphism class contains a unique representative (O, α 0 ) such that Tr(α 0 ) ∈ {0, .…”

“…As shown by Bhargava and the fourth-named author in [14], if O is a cubic order and I ∈ I(O) [2] (resp., I ∈ I(O) [2]) is an ideal of O whose class in Cl(O) is 2-torsion, then I corresponds via Bhargava's parametrization to a reducible (resp., irreducible) integral orbit of a certain coregular representation. 2 In [39, Theorem 9], the second-named author applied a higher-degree generalization of Bhargava's parametrization due to Wood (see §2.2) in conjunction with the aforementioned general approach for counting irreducible integral orbits to determine the average size of the quantity (1) # Cl(O) [2] − 2 1− n+r 1 2…”

“…Shintani's result was later recovered using an elementary geometry-of-numbers argument by Bhargava and the fourthnamed author (see [15, §4.1.2]), who applied it to determine the average size of the 3-torsion in class groups of quadratic orders. Secondly, in [1], Altug, the first-and fourth-named authors, and Wilson used both L-function techniques and geometry-of-numbers methods to prove an asymptotic for the number of D 4 -reducible orbits of GL 2 × SL 3 acting on pairs of integer-matrix ternary quadratic forms. 3 However, all of these works rely crucially on situation-specific techniques and on the fact that the representations under consideration are of relatively small dimension.…”