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Inductive methods for counting number fields
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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.
Forward citations
Cited by 3 Pith papers
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Malle's Conjecture for Galois octic fields over $\mathbb Q$
The number of octic D4-fields with absolute discriminant below X is asymptotic to an explicit constant times X^{1/4} (log X)^2.
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A refined Malle conjecture for Heisenberg groups
The leading constant for Malle's conjecture for Heis_4-extensions of Q decomposes as a sum of two Euler products due to a transcendental Brauer–Manin obstruction, yielding the first discriminant-ordering failure of lo...
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Counting abelian number fields with restricted ramification type
For finite abelian G, G-extensions of bounded height with restricted tame ramification type satisfy an explicit Malle-type asymptotic whose constant is governed by a partially unramified Brauer group.
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