1 results
Malle's conjecture for
$S_n\times A$ for 
$n = 3,4,5$
- Part of
-
- Journal:
- Compositio Mathematica / Volume 157 / Issue 1 / January 2021
- Published online by Cambridge University Press:
- 15 February 2021, pp. 83-121
- Print publication:
- January 2021
-
- Article
- Export citation
-
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.
