Conditional on the Loughran-Santens conjecture, the count of Galois Heis_4-extensions of Q with |disc|≤X is predicted to grow like ½(C0+Cβ) X^{1/32}(log X)^8, with the constant a sum of two Euler products.
[AO21] Brandon Alberts and Evan O’Dorney
3 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group $G$. Our method relies on having asymptotic counts for $T$-extensions for some normal subgroup $T$ of $G$, uniform bounds for the number of such $T$-extensions, and possibly weak bounds on the asymptotic number of $G/T$-extensions. However, we do not require that most $T$-extensions of a $G/T$-extension are $G$-extensions. Our new results use $T$ either abelian or $S_3^m$, though our framework is general.
fields
math.NT 3years
2026 3representative citing papers
Authors conjecture an explicit leading constant for the number of number fields of bounded discriminant by transferring Manin philosophy to classifying stacks, plus related conjectures on multi-heights and local conditions.
Develops multiple Dirichlet series methods to count G-extensions for infinitely many new Galois groups G, with unconditional results for concentrated groups and conditional asymptotics including all nilpotency class 2 groups.
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A refined Malle conjecture for Heisenberg groups
Conditional on the Loughran-Santens conjecture, the count of Galois Heis_4-extensions of Q with |disc|≤X is predicted to grow like ½(C0+Cβ) X^{1/32}(log X)^8, with the constant a sum of two Euler products.
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The leading constant in Malle's conjecture
Authors conjecture an explicit leading constant for the number of number fields of bounded discriminant by transferring Manin philosophy to classifying stacks, plus related conjectures on multi-heights and local conditions.
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Develops multiple Dirichlet series methods to count G-extensions for infinitely many new Galois groups G, with unconditional results for concentrated groups and conditional asymptotics including all nilpotency class 2 groups.