Math Conjectures

Weak Malle Discriminant-Growth Conjecture

OPENLandmarkConjectureProposed 2002 · Standard version

Canonical statement

Let kk be a number field and G≤SnG\le S_n a transitive permutation group. Let Nk,G(X)N_{k,G}(X) count kk-isomorphism classes of degree-nn extensions L/kL/k whose Galois closure has permutation group GG and whose relative discriminant has absolute norm at most XX. Define
ind⁡(g)=n−#{orbits of g},a(G)=(min⁡1≠g∈Gind⁡(g))−1. \operatorname{ind}(g)=n-\#\{\text{orbits of }g\},\qquad a(G)=\left(\min_{1\ne g\in G}\operatorname{ind}(g)\right)^{-1}.
For every ε>0\varepsilon>0,
Xa(G)≪k,GNk,G(X)≪k,G,εXa(G)+ε X^{a(G)}\ll_{k,G}N_{k,G}(X)\ll_{k,G,\varepsilon}X^{a(G)+\varepsilon}
as X→∞X\to\infty.
View source LaTeX
Let \(k\) be a number field and \(G\le S_n\) a transitive permutation group. Let \(N_{k,G}(X)\) count \(k\)-isomorphism classes of degree-\(n\) extensions \(L/k\) whose Galois closure has permutation group \(G\) and whose relative discriminant has absolute norm at most \(X\). Define
\[
 \operatorname{ind}(g)=n-\#\{\text{orbits of }g\},\qquad
 a(G)=\left(\min_{1\ne g\in G}\operatorname{ind}(g)\right)^{-1}.
\]
For every \(\varepsilon>0\),
\[
 X^{a(G)}\ll_{k,G}N_{k,G}(X)\ll_{k,G,\varepsilon}X^{a(G)+\varepsilon}
\]
as \(X\to\infty\).
NT-GALOIS-041:[Malle2002Distribution][Kluners2005MalleCounterexample][Wang2025MalleCorrection]
Number theoryAlgebra and representation theory

Notes

For a transitive permutation group G≤SnG\le S_n, Malle predicted the growth of degree-nn number-field extensions with Galois closure group GG, ordered by discriminant. The basic exponent a(G)a(G) is the reciprocal of the smallest index of a nonidentity element, and the robust conjecture says that the counting function lies between constant multiples of Xa(G)X^{a(G)} and Xa(G)+εX^{a(G)+\varepsilon} [Malle2002Distribution].

Malle also proposed a sharper asymptotic with a specific logarithmic exponent. Klüners found a counterexample to that strong formulation [Kluners2005MalleCounterexample]. Türkelli proposed a correction based on cyclotomic intersections, but further counterexamples and a more finely stratified replacement appeared in 2025 [Wang2025MalleCorrection]. None of these counterexamples refutes the underlying power exponent recorded here.

The exponent prediction is known for numerous abelian, nilpotent, symmetric, wreath-product, and low-degree cases, but not for every transitive group. The entry intentionally stops at the two-sided exponent estimate: there is not yet a universally accepted corrected logarithmic exponent and leading constant suitable for a canonical card.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.