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.
We describe the relations among the
ℓ
\ell
-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the
ℓ
\ell
-torsion conjecture for
ℓ
\ell
-groups and the other two conjectures for nilpotent groups.
Elementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.
Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of
$\mathbb {Q}$
and to give a new proof of the vanishing of Massey triple products in Galois cohomology.
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