Malle proposed a conjecture for counting the number of
-extensions
with discriminant
bounded above by
,
denoted
, where
is a fixed transitive
subgroup
and
tends towards infinity. We introduce a refinement of Malle’s conjecture, if
is a group
with a nontrivial Galois action then we consider the set of crossed homomorphisms in
(or equivalently
-coclasses
in
)
with bounded discriminant. This has a natural interpretation given by counting
-extensions
for some fixed
and prescribed
extension class
.
If
is an abelian group with any Galois action, we compute the
asymptotic growth rate of this refined counting function for
(and
equivalently for
)
and show that it is a natural generalization of Malle’s conjecture. The proof
technique is in essence an application of a theorem of Wiles on generalized Selmer
groups, and additionally gives the asymptotic main term when restricted to certain
local behaviors. As a consequence, whenever the inverse Galois problem is solved for
over
and
has an abelian normal
subgroup
we prove a
nontrivial lower bound for
given by a nonzero power of
times a power of
.
For many groups, including many solvable groups, these are the first known
nontrivial lower bounds. These bounds prove Malle’s predicted lower bounds for a
large family of groups, and for an infinite subfamily they generalize Klüners’
counterexample to Malle’s conjecture and verify the corrected lower bounds predicted
by Türkelli.
