Malle’s conjecture on discriminant growth of extensions with specified Galois group

Establish that for a number field k and a finite transitive permutation group G ≤ S_n, the counting function Z(k,G;X) = #{ K/k : Gal(K/k) = G and N_{k/ℚ}(D(K/k)) ≤ X } satisfies the asymptotic Z(k,G;X) ~ c(k,G) X^{a(G)} log(X)^{b(k,G)-1} as X → ∞ for some constant c(k,G) > 0, where a(G) and b(k,G) are the explicit exponents predicted by Malle’s conjecture.

Background

The paper recalls Malle’s conjecture in the number field setting as a central guiding problem for counting field extensions with specified Galois group by discriminant. The conjecture predicts a precise asymptotic with explicit exponents a(G) and b(k,G).

Within the broader context, many tame cases over global function fields are known or proved under additional hypotheses, while the conjecture in full generality remains open. The authors focus on local function fields of characteristic p and groups with p dividing |G|, where the original exponent a(G) is known to be incorrect, motivating refined investigations in positive characteristic.