Weak Form of Malle’s Conjecture for Pushforward Discriminants

Prove that for every number field k, every finite permutation group G with normal subgroup T ◁ G, and the quotient map q: G → G/T, the number of surjections π ∈ Sur(G_k,G/T) ordered by the pushforward discriminant q_*disc satisfies # { π ∈ Sur(G_k,G/T) : |q_*disc(π)| ≤ X } ≪_ε X^{1/a(G\setminus T) + ε}, where a(G\setminus T) = min_{g ∈ G\setminus T} ind(g).

Background

To control the image q_*Sur(G_k,G;X) and bound contributions from fibers, the paper introduces the pushforward discriminant q_*disc on G/T-extensions: it is defined locally by minimizing the conductor exponent across lifts to G and assembled globally as an ideal. This creates an ordering on G/T-extensions tailored to the lifting problem.

The conjecture is a pushforward analogue of Malle’s weak upper bound, heuristically supported by the Ellenberg–Venkatesh philosophy. It is known in certain nilpotent cases (via results on discriminant multiplicity), but remains open in general. Proving it would directly feed the inductive counting framework developed in the paper.