Twisted Number Field Counting Conjecture

Establish, for a number field k, a transitive permutation group G of degree n, a proper normal subgroup T ◁ G with quotient map q: G → G/T, and any π ∈ q_*Sur(G_k,G), an asymptotic formula for the size of the fiber {ψ ∈ q_*^{-1}(π) : |disc_G(ψ)| ≤ X} of the form c X^{1/a} (log X)^{b-1} as X → ∞, for some positive constants a, b, c depending on k, G, T, and π.

Background

To inductively count G-extensions, the paper partitions extensions by the Galois subfield fixed by a chosen normal subgroup T ◁ G. This leads to studying fibers of the pushforward map q_*: Sur(G_k,G) → Sur(G_k,G/T), where each fiber corresponds to extensions whose Galois closure restricts to a fixed G/T-quotient.

The twisted conjecture refines Malle’s Conjecture to each fiber. Its parameters a and b are predicted in analogy with Malle’s exponents, and its resolution (with explicit uniformity in π) enables summing the fibers to deduce global asymptotics for #F_{n,k}(G;X). The authors prove new cases when T is abelian or when T is S_3^m in certain wreath product settings, but the conjecture is open in full generality.