Lower bounds for counting $A_4$-quartic fields

TL;DR

This work proves a nontrivial lower bound for the number of A4-quartic fields L/ℚ with Δ_L ≤ X, achieving # {L} ≫ X^{1/2} log X and thereby matching the predicted order from Malle’s conjecture in the balanced quartic setting. The authors develop an integrated approach combining (i) counting ideals of squarefree norm with class-group constraints, (ii) establishing an averaged residue formula for cyclic cubic zeta functions via Dirichlet L-functions and a bilinear sieve, and (iii) translating A4-quartic counts into cubic-resolvent data through the parametrization of MR3215550, including a careful Selmer-group analysis. A key technical ingredient is the averaged bound for ζ_F^*(1) over cyclic cubic fields unramified at 3, which yields an explicit main term involving constants α and β_d; summing over cubic resolvents then produces the global X^{1/2} log X lower bound. The results connect to the stack-theoretic predictions of Manin–Ellenberg–Satriano–Zurieck-Brown, showing agreement with the Tamagawa-measure framework up to residual local factors, and offer techniques potentially adaptable to other inductive approaches to Malle’s conjecture and related counting problems in arithmetic statistics.

Abstract

A conjecture of Malle predicts the quantity of number fields with bounded discriminant of given Galois group. We present a lower bound matching this in the case of quartic fields with Galois group $A_4$.

Theorems & Definitions (55)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 7
• Lemma 8
• Definition 9
• Lemma 10