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Malle's conjecture for $S_n\times A$ for $n = 3,4,5$

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  15 February 2021

Jiuya Wang
Jiuya Wang*
Affiliation:
Department of Mathematics, Duke University, 120 Science Drive, 117 Physics Building, Durham, NC27708, [email protected]
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Abstract

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

Keywords

Malle's conjecturecompositumuniformity estimatecounter exampledensity of discriminants

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants 11R45: Density theorems
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 1 , January 2021 , pp. 83 - 121
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Copyright
© The Author(s) 2021

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