$\ell$-torsion bounds for the class group of number fields with an $\ell$-group as Galois group
Jürgen Klüners Jiuya Wang
math.NT 11R37 11Y40


ℓ-torsion bounds]ℓ-torsion bounds for the class group of number fields with an  ℓ-group as Galois group


 Universität Paderborn, Institut für 
  Mathematik, Warburger Str. 100, 33098 Paderborn, Germany

klueners@math.uni-paderborn.de


Duke University, Department of Mathematics, Durham, NC, 27708, US
wangjiuy@math.duke.edu
[2010]Primary 11R29; Secondary 11R37
 
	We describe the relations among the ℓ-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-torsion conjecture for ℓ-groups and the other two conjectures for nilpotent groups.



[
    Jiuya Wang
    December 30, 2023
=====================





§ INTRODUCTION


In this paper, we study several conjectures in arithmetic statistics. In all situations E/F will be an extension of degree n of number fields with absolute discriminant (E/F) which is the absolute norm of the discriminant ideal (E/F).


	 Let ℓ be a prime number and E/F be an extension with absolute discriminant D and degree n = [E:F]. Then the size h_ℓ(E) of the ℓ-torsion in the class group of E is
	
    h_ℓ(E)= O_ϵ,n,ℓ,F(D^ϵ)ϵ>0.


We write (E/F)=G ≤ S_n if the Galois group of the Galois closure Ê/F of E/F viewed as
permutation group on the set of embeddings of E into Ê is permutation isomorphic to G.
In our second conjecture we consider the function

    N(F,G;X):=#{E/F|(E/F)=G, (E/F)≤ X }.
  By a famous result of Hermite (e.g. see <cit.>) there are only finitely many number fields with the same given discriminant. 


	Let G≤ S_n be a transitive group acting on Ω={1,…,n}.
	
		
  * For g∈ G we define the index (g):= n- Ω.
		
  * For n>1 let a(G):=(G):=min{(g): 𝕀 g∈ G}.
	

Note that a(G) here is the inverse of the a(G) defined in <cit.>.

	Let F be a number field and  G≤ S_n be a transitive group. Then we have
	
    N(F,G;X)=O_ϵ,F(X^1/a(G)+ϵ) ϵ>0.


In the same paper Malle conjectures a lower bound which is equivalent to 
    lim inf_X→∞ X^-1/a(G)N(F,G;X)>0.

We remark that Malle gives a refined version of the conjecture in <cit.>, which we do not need in our context here. There are also counter-examples known due to the first author <cit.> for this refined conjecture.

In our last conjecture we consider the number

    a_D:=#{E/F|(E/F)=G, (E/F)=D }

 of G-extensions of F with discriminant (E/F)=D.

	Let F be a number field and G≤ S_n be a transitive group. Then for all D∈ we have
		
    a_D = O_ϵ,F,n(D^ϵ)>0.



The goal of this paper is twofold: on one hand, to show the relations among Conjectures <ref>, <ref>, and <ref>; on the other hand, to show that these conjectures have affirmative answers when we restrict our discussion to certain general families of number fields. 

These conjectures share the common feature that they are all about giving upper bounds on the number of arithmetic objects, including number fields and class numbers. We will show that they are almost all equivalent to each other.

	Let F be a number field. Then
	
		
  * Assume that Conjecture <ref> is true for all G≤ S_nℓ, then Conjecture <ref> is true for ℓ and for all extensions E/F of degree n.
		
  * Assume that Conjecture <ref> is true for all solvable extensions E/F and all prime numbers ℓ. Then Conjecture <ref> is true for all solvable groups G.
	
 

	Part (i) is shown in <cit.> and <cit.>.  The second part is <cit.>.

  A little bit stronger Alberts proves in <cit.> that Conjecture <ref> is true for solvable groups, if we assume that the torsion conjecture is true in average.   

Noticing that Conjecture <ref> is essentially an average statement of  Conjecture <ref>, it is not surprising that in general Conjecture <ref> implies Conjecture <ref>. We manage to prove that they are equivalent.

	Let F be a number field. Then 


  * Conjecture <ref> for all finite groups G implies Conjecture <ref> for all finite groups G.

  * Conjecture <ref> for G≤ S_n implies 
Conjecture <ref> for G.
	

The first part is shown in Theorem <ref> and the second part in Theorem <ref>.


We remark that Proposition <ref> (i) and Theorem <ref> (i) will be also true for the class of solvable extensions, see Corollary <ref>. Therefore we get: 

	Conjectures <ref>, <ref>, and <ref> are equivalent when we restrict to solvable extensions E/F.

We remark that we do not expect  that the similar statement for nilpotent extensions E/F is true. The reason is that if you want to consider ℓ–torsion  of the class group for p–extensions, then the resulting Galois groups are solvable, but in most cases not nilpotent.



We then focus on proving some special cases of these conjectures. Conjecture <ref> has a clear dichotomy depending on whether _E[ℓ] is randomly distributed. Classically, it is only known to be true when F= and ℓ = 2 for [E:F] = 2 by Gauss using genus theory. Firstly, we give a compact proof on all cases for Conjecture <ref> where a similar argument with ℓ=2 for quadratic extensions applies, i.e., when the distribution of _E[ℓ] is governed by genus theory. We say a field extension E/F is an ℓ-extension when  (E/F) is an ℓ-group. 

	Conjecture <ref> holds for the ℓ-torsion of class groups of ℓ-extensions.
	
We will prove a more precise version in Theorem <ref>.

 
 Results on Conjecture <ref> for ℓ, where E/F is not an ℓ–extension are much more difficult to prove. 
 Heuristically, Conjecture <ref> is shown to be a consequence of the moments version of the Cohen-Lenstra heuristics in <cit.>. We mention some results in this direction: ℓ = 2 <cit.>, ℓ = 3 with d≤ 4 <cit.>, and for arbitrary ℓ with (E/F)= (/p)^r (r>1) <cit.>. There are also recent works <cit.> on proving a non-trivial bound on the ℓ-torsion of class groups for almost all number fields in some certain family of number fields. 

In Section <ref> we prove the following theorem by applying Theorem <ref>. 

	Conjecture <ref> holds for nilpotent number field extensions E/F.

Finally, by applying Theorems <ref> and <ref>, we recover the following theorem.
  
	Conjecture <ref> holds for nilpotent number field extensions E/F. 

We mention that Theorem <ref> is also proved in <cit.> for Galois nilpotent extensions and in <cit.> for general nilpotent extensions. We recover this theorem as a direct consequence of Theorem <ref>, and therefore give a short and simplified proof.

We finally remark that, as shown in the proof of Theorem <ref>, the fundamental reason for these conjectures to hold in such a perfect shape in these cases is completely group theoretic, i.e., nilpotent groups have non-trivial center. 


All results in this paper are effective.









































§ ℓ-TORSION CONJECTURE

In this section we prove Theorem <ref>. We give a more detailed version in   Theorem <ref>. 
We start with the following theorem, which is proved in <cit.> for odd ℓ and generalized in <cit.> to ℓ=2.  In order to keep this
note self-contained we give a proof of this statement here. In the following we use the notion places for finite prime ideals and infinite places.

  Let E/F be a cyclic extension of number fields of degree ℓ ramified in t places. Let e:=max(t,1). Then





     
    _ℓ(_E)≤ℓ(e-1 + _ℓ(_F)).
  


	Firstly, the Galois group (E/F) = ⟨σ⟩
        acts on the _ℓ-vector space _E[ℓ]. Since
        σ^ℓ = id, the minimal polynomial for
        σ on _E[ℓ] divides x^ℓ-1 = (x-1)^ℓ.
        Therefore, the only eigenvalue is 1 and each
        Jordan block has at most size ℓ. It suffices to prove that
        the number of Jordan blocks is bounded by
        e-1+_ℓ(_F).
	
	The number of Jordan blocks is equal to the dimension of the
        maximal quotient space on which σ acts trivially.



Denote the corresponding class field by M. Notice that since σ acts trivially, the field M/F is Galois and abelian, therefore (M/E)≅ C_ℓ^s for some s≥ 0 and (M/F)≅ C_ℓ^s+1 or
(M/F)≅ C_ℓ^2× C_ℓ^s-1. We would like to prove that
s≤ e-1+_ℓ(_F). We note that the second case can only happen
when t=0, i.e. E/F is unramified. In this case we see s=_ℓ(_F)
and our claim is proved.

In the first case denote by L/F the maximal unramified (including infinite places) subextension
of M/F. We know by construction that (L/F)≅
C_ℓ^_ℓ(_F).  M/L is abelian and it has no subextension which is everywhere unramified (including infinite places).
Therefore (M/L) is generated by the inertia groups of the ramified
prime ideals including the infinite ones. Let  be a prime ideal of _F which is ramified
in M. Since M/E is unramified, we see that the inertia group has
size ℓ and is therefore cyclic. The same applies for the prime
ideal in L lying above . The inertia groups at infinite places are always cyclic and we see that each ramified prime in
E/F can increase the rank of (M/L) by at most 1. We therefore
get:

    _ℓ((M/F)) ≤_ℓ(_F) + e _ℓ((M/E))=_ℓ((M/F))-1.















 
	In case  _ℓ (_F)=0 <cit.> and <cit.> prove a slightly better upper bound for _ℓ(_E), i.e. 
	_ℓ(_E) ≤ (ℓ-1) · (e-1). 
	It is known that for F= and ℓ=2 this bound is sharp by genus theory.
	Furthermore, <cit.> also gives a better bound for the cyclic of order ℓ^r-case compared to the inductive approach we present in Theorem <ref>.

We do not prove this remark since we are only interested in the asymptotic behavior and therefore the change of constants does not matter. 	
In order to prove Theorem <ref> for non normal extensions we need the following lemma.	
 
	 Let n=ℓ^r and G≤ S_n be an ℓ-group and E/F be an extension of number fields with (E/F)≅ G. Then there exists a tower of fields
	 
    F=F_0 ≤ F_1≤…≤ F_r-1≤ F_r=E

	 such that (F_i+1/F_i)=C_ℓ for all 0≤ i ≤ r-1.


Let Ẽ be the normal closure of E over F. Denote H to be the subgroup of G fixing E and choose a maximal subgroup G_1≤ G that contains H. Note that all maximal subgroups of an ℓ-group have index ℓ and are normal. Define F_1 to be the subfield of Ẽ fixed by G_1. Inductively, we can find a sequence of subgroups G≅ G_0⊃ G_1⊃⋯⊃ G_r = H with [G_i:G_i+1]= ℓ for every 0≤ i≤ r-1 and define F_i to be the subfield fixed by G_i. 








	
 Now we prove our main result of this section. We remark that <cit.> describes just before Theorem 3 how to get this result for normal ℓ-extensions.


	Let n=ℓ^r, G≤ S_n be a transitive ℓ-group, and E/F be an ex­ten­sion of number fields with (E/F)≅ G, 
  and with tower as defined in (<ref>).
  Let t_i be the number of ramified places in F_i+1/F_i and 
  e_i:=max(t_i,1). Then we get:







    
    _ℓ(_E)≤∑_i=0^r-1ℓ^r-i(e_i-1) + n_ℓ(_F).



By Lemma <ref> we find a tower of cyclic extensions of order ℓ.
The assertion now follows by applying Theorem <ref> for each step.	
	
The above version is still a little bit complicated since we need to
know the number of ramified places in each step. It would be
much nicer to have a bound which is only depending on the number of
ramified places of F. Let  be a prime
ideal of _F  which ramifies for the first time
in F_i+1. We have the extreme case if  splits completely in F_i
which means that there are ℓ^i places over  lying in F_i.

Therefore, if t is the number of prime ideals in _F which are
ramified in E/F, then we get that t_i≤ e_i ≤max(ℓ^i t,1) in
(<ref>). Therefore we get:

  Let G≤ S_n be a transitive ℓ-group and E/F be an extension with (E/F)≅ G of degree n=ℓ^r which is
   ramified in t places. Then
   
    _ℓ(_E)≤  r n t + n _ℓ(_F).

   
   
   


  Note that e_i-1 ≤ℓ^i t and using this in (<ref>) we get
  _ℓ(_E)≤  r n t + n_ℓ(_F)
  and the assertion follows easily.



In the next step we would like to know an upper bound for the number
of different prime ideals dividing the discriminant of E/F. We use
the following standard result, e.g. see <cit.>.

  For an integer n we denote by ω(n) the number of distinct
  prime factors. Then there exists an explicit constant C>0 such that for every n> 2, 
  
    ω(n)≤ Clog n/loglog n.

  Let F be a number field of degree d.
  Then for an integral ideal 𝔫 O_F with absolute norm n = |𝔫|>2, the number ω(𝔫) of distinct prime ideal factors is bounded by
  
    ω(𝔫) ≤ d ·ω(n) ≤ Cdlog n/loglog n.


We remark that the average order of ω(n) is loglog n. Note that we can choose C=1.3841, see
<cit.>.

Using that the number of ramified prime ideals in a relative extension E/F is ω((E/F)), we prove our main result by applying Proposition <ref> and Lemma <ref>. 

  Let E/F be an ℓ-group extension of degree n=ℓ^r and absolute discriminant
  D:=(E/F), and define d:=[F:]. Then we get:
  
    _ℓ(_E) ≤ n _ℓ(_F) + nr· C d log D/loglog D for D>2,

 equivalently, we get for the size h_ℓ(E) of the ℓ-torsion part _E[ℓ]:
    
    h_ℓ(E) ≤  h_ℓ(F)^n · D^Cndrlogℓ/loglog D = O_ϵ,F, n(D^)
      >0.

  For D=1 we get _ℓ(_E) ≤ n _ℓ(_F) and therefore h_ℓ(E) ≤  h_ℓ(F)^n.
 
Note that the case D=2 is not possible by Hilbert's ramification theory. The reader can also find an independent proof of Theorem <ref> by G. Gras <cit.>, which he gave after seeing our preprint. We remark that the Galois group of all fields in his family ℱ_K^p^e are also p–groups. 




 
 	Note that we easily get the following estimate for the ℓ^s-torsion 
    h_ℓ^s(E) ≤ h_ℓ(E)^s ≤ h_ℓ(F)^ns· D^Cndrslogℓ/loglog D =  O_ϵ,F,n,s(D^ϵ).

 


Theorem <ref> or Lemma <ref> are not expected to be sharp, but will be sufficient for our purpose, since we only aim at proving Conjecture <ref>. However, it is also an independent interesting question to study the upper bound at a finer scale. We mention results along this direction <cit.> for F=, where a sharp upper bound is obtained for ℓ=2 and certain special family of multi-quadratic number fields. 












§ DISCRIMINANT MULTIPLICITY CONJECTURE FOR NILPOTENT EXTENSIONS

The goal of this section is to prove Theorem <ref>  which answers Conjecture <ref> positively in the nilpotent case. As usual we can reduce the nilpotent case to the ℓ–group case, but we have to be a little bit careful (see Lemma <ref>) that this reduction is compatible with permutation groups. For the ℓ–group case we use the upper bound proved in Theorems <ref> and <ref>. 





In a first step we prove this theorem for ℓ-groups and then use this for proving the case of arbitrary nilpotent groups. For the latter step
 we need a group theoretic lemma. This states that any transitive nilpotent permutation group G≤ S_n is isomorphic to a natural direct product of its ℓ-Sylow subgroups. It is a standard fact that all nilpotent groups are isomorphic to the direct product of their ℓ-Sylow subgroups, however we emphasize that we need to prove the isomorphism in the category of permutation groups. Equivalently, this means that all nilpotent G-extensions (not necessarily Galois) can be realized as a compositum of ℓ-extensions. 


	A transitive nilpotent permutation group G≤ S_n is permutation isomorphic to the natural direct product of transitive permutation groups G_ℓ≤ S_n_ℓ,
	
    G ≃∏_ℓ G_ℓ n=∏_ℓ n_ℓ,

	where the G_ℓ are isomorphic to the ℓ-Sylow subgroups of G and n_ℓ is the maximal ℓ-power dividing n.


	Firstly, it is a standard result that a nilpotent group G is equal to the direct product ∏_ℓ| n_ℓ(G) where _ℓ(G) is the ℓ-Sylow subgroup of G. Let H≤ G be a stabilizer of a point. This means that G can be realized by the action of G on the left cosets of G/H.
	Now H is nilpotent as well and therefore it is a direct product of its Sylow subgroups. We get:
	
    H=∏_ℓ| n_ℓ(H)  with _ℓ(H) ≤_ℓ(G).

	Now we can define the permutation groups G_ℓ by the action of _ℓ(G) on the left cosets of _ℓ(G)/_ℓ(H) for each prime ℓ dividing n.
	Since H=∏_ℓ| n_ℓ(H) we see that G is permutation isomorphic to the natural direct product ∏_ℓ| n G_ℓ≤ S_n of permutation groups. 












	Let F be a number field of degree d, ℓ be a prime number, and  be an ideal of _F. Then the number of C_ℓ-extensions E/F with (E/F)=
	is bounded above by
	
    O_d, ℓ(h_ℓ(F)·ℓ^ω())=O_ϵ, d, ℓ( h_ℓ(F) · D^ϵ) = O_ϵ, F,ℓ( D^ϵ) >0.
   


	Let E/F be such an extension. Then by class field theory the finite part _0 of the conductor has the property that =_0^ℓ-1. Denote
	by _∞ the set of real places, if ℓ=2 and let _∞=∅ otherwise.
	Then E is contained in the ray class field of =_0_∞ and we need an upper bound on the size of the ℓ-torsion of this ray class group _.
	By class field theory we have the following short exact sequence: 

    ∏_|_0 (_F/^e_)^* ×∏_|_∞ C_2 →_→_F → 0  _0=∏_|_0^e_.

Therefore the ℓ-rank of _ is bounded above by ∑_|_0_ℓ((_F/^e_)^*) + d + _ℓ(_F). Note that
_ℓ((_F/^e_)^*) ≤ 1 for ℓ∉ and _ℓ((_F/^e_)^*) ≤ [F_:_ℓ]+1, if ℓ∈.
Considering the wildly ramified primes, the extreme case happens when all wildly ramified primes are dividing _0 and ℓ splits in F. In this
situation the wildly ramified primes might increase the ℓ-rank by 2d. The infinite places might increase the 2-rank by d and therefore we get the following upper bound:

    _ℓ(_)≤ω(_0)+3d+_ℓ(_F).


Therefore the number of C_ℓ-extensions E/F with (E/F) = is bounded by |_[ℓ]|=O_d, ℓ(h_ℓ(F)·ℓ^ω())=O_ϵ, d, ℓ( h_ℓ(F) · D^ϵ) = O_ϵ, F,ℓ( D^ϵ) for all >0 by Proposition <ref>. 
	



	Denote by b_D the number of ideals  of _F such that ||=D. We claim that b_D= O_d(C^ω(D)) for some C depending on the degree d. This is bounded by O_ϵ, d(D^ϵ). Therefore it suffices to prove that the number of G-extensions E/F with (E/F) = is bounded by O_ϵ, F, n(D^ϵ).
	
	In order to prove the claim note that b_D is multiplicative and therefore it suffices to prove it for prime powers D=p^s. The worst case happens
	when p is split in F. 

	Then the number of ideals is equal to d+s-1d-1≤ (s+1)^d-1 and s=O_d(log D) which gives d+s-1d-1 = O_d((log D)^d-1)=O_ϵ, d(D^ϵ) for all >0.
		
	
	 
	Let us assume that G≤ S_n is a transitive ℓ-group of degree n = ℓ^r. We proceed by induction on r. When r = 1, then G=C_ℓ and we apply Lemma <ref>.
	
	Suppose the statement holds for ℓ-extensions of degree n = ℓ^r. Given an arbitrary ℓ-extension E/F of degree ℓ^r+1, there is a chain of subfields E = E_r+1≥ E_r≥⋯≥ E_0 = F using Lemma <ref>. 
	Denote (E_r/F)=, then we have ^ℓ·' = by the discriminant formula for towers. 
	
	By induction for n=ℓ^r, the number of extensions E_r/F with (E_r/F) = | is bounded by O_ϵ, F, n( ||^ϵ ). Using Lemma <ref> the number of E_r+1/E_r with relative discriminant _E_r/F((E_r+1/E_r)) = ' is bounded by O_ϵ, d, ℓ (h_ℓ(E_r) · |'|^ϵ ). Since h_ℓ(E_r)=O_,F,n(||^) by Theorem  <ref>, we get the bound O_ϵ, F, n( |'|^ϵ ||^ϵ) for the number of E_r+1/E_r with relative discriminant _E_r/F((E_r+1/E_r)) = '.
	
	  Therefore for each ^ℓ|, the number of extensions E_r+1/F with (E_r+1/F)= and (E_r/F)= is bounded by O_ϵ, d, n (D^ϵ). The number of divisors | is bounded by O_ϵ(D^ϵ). So the number of E_r+1/F with (E_r+1/F) = in total is bounded by O_ϵ, F, n( D^ϵ). This finishes the proof of the discriminant multiplicity conjecture for general ℓ-extensions. 
	
	Secondly, we deduce the discriminant multiplicity conjecture for nilpotent extensions from the one for ℓ-extensions. 
	Given a transitive nilpotent permutation group G≤ S_n, by Lemma <ref>, we have 
	
    G ≃∏_i=1^s G_ℓ_i≤∏_i=1^s S_ℓ_i^s_i≤ S_n n=∏_i=1^s ℓ_i^s_i.
  Therefore each G-extension E/F is the compositum of ℓ_i-extensions E_ℓ_i/F with (E_ℓ_i/F) = G_ℓ≤ S_ℓ_i^s_i. Therefore the number of G-extensions E/F with (E/F)= is bounded by the number of tuples (E_ℓ_1,…,E_l_s) of ℓ_i-extensions with (E_ℓ_i/F) | and (E_ℓ_i/F) = G_ℓ_i for each i. Combining the result on ℓ-extensions and the fact that the number of divisors of 

	 is bounded by O_ϵ, n(D^ϵ), we deduce that the number of E_ℓ/F for each prime ℓ| n is bounded by O_ϵ, F, n_ℓ( D^ϵ). Taking the product over all ℓ| n, we get that the number of such tuples is bounded by 
	 O_ϵ, F, n( D^ω(n)ϵ) = O_ϵ, F, n(D^ϵ).




§ MALLE'S CONJECTURE FOR NILPOTENT EXTENSIONS

The goal of this section is to prove Theorems <ref> and <ref>. The proof of Theorem <ref> is split into proving Theorems <ref> and <ref>. The proof of Theorem <ref> is the content of Corollary <ref>.

	Let G≤ S_n be a transitive permutation group  and F be a number field. Assume that Conjecture <ref> is true for F and G. Then Conjecture <ref> is also true for F and G, i.e. the number of G-extensions E/F with (E/F)≤ X is bounded above by:
	
    N(F,G; X) = O_F, ϵ,n( X^1/a(G)+ϵ) >0.



	Let 𝒜 : = { D∈ℤ : p | D  p^a(G)| D  } and 𝒜(X):= ♯{ D∈𝒜 : D<X } be the counting function associated to 𝒜. Let a_D be the number of fields E/F with Galois group G and D=(E/F). Note that a_D=0, if D∉𝒜, see <cit.>. Then for any ϵ>0, there exists a constant C_F, ϵ,n 
	by Conjecture <ref> such that
	
    N(F, G;X) = ∑_D∈𝒜,D<X  a_D ≤∑_D∈𝒜,D<X C_F, ϵ,n· D^ϵ≤ C_F, ϵ,n·𝒜(X)· X^ϵ.

    We will show that
	
    𝒜(X) ∼ C'· X^1/a(G),

	for some C'>0, therefore 𝒜(X) = O(X^1/a(G)). The generating series f(s) of 𝒜 is 
	
    f(s) = ∏_p (1+ ∑_k≥ a(G) p^-ks) = ζ(a(G)s) · g(s),

	where
	
    g(s) = ∏_p (1+ ∑_a(G)+1 ≤ k≤ 2a(G)-1 p^-ks),

   is a holomorphic function when (s)> 1/(a(G)+1). The function f(s) thus has an analytic continuation to (s)≥ 1/a(G) except for a simple pole at s = 1/a(G). We get (<ref>) from a Tauberian theorem, e.g. see <cit.>. Then the result follows since C_F, ϵ,n·𝒜(X) · X^ϵ = O_F,ϵ,n(X^1/a(G)+ϵ). 		









	Note in the above proof that 𝒜(X)=O_ϵ(X^1/a(G)+ϵ) for all ϵ>0 can be easily derived without using a Tauberian theorem. This is certainly sufficient to finish the proof.
Using Theorems <ref> and <ref> we immediately get the following corollary which is the content of Theorem <ref>.

Let G be a transitive nilpotent group. Then Conjecture <ref> is true for all ϵ>0 and all number fields F.






Now we prove some version of the inverse implication of Theorem <ref>.
We cannot expect that Conjecture <ref> for a single G implies Conjecture <ref> for the same G, because it is easy to write down series of numbers a_m which have a given asymptotic behavior, but a single a_m is not bounded by O(m^ϵ) for all ϵ>0. E.g. we can take the series 

    a_m = √(m)   if m is a square,
     0    otherwise. and we see ∑_m≤ x a_m ∼ x/2,

but a_m=√(m) for infinitely many m. In order to prove Conjecture <ref> we need some control over the higher moments. We can get this control when we assume Conjecture <ref> for so-called allowable permutation groups (see Definition <ref>), which are suitable subgroups of the direct product G^k. Using this assumption we are able to prove in Theorem <ref> that Conjecture <ref> holds. 

 Let us consider G-extensions E_i/F for 1≤ i≤ k for a transitive G≤ S_n. 
 We are interested in the Galois group of the tensor product E= ⊗_i=1^k E_i which is a subgroup of G^k. Note that K=K_1⋯ K_m is a product of fields and for m=1 it can be interpreted as the compositum of the E_i. We define ρ: G_F→ G^k ≤ S_n^k, where G_F denotes the absolute Galois group of F. The image ρ(G_F) might not be transitive of degree n^k, equivalently, this means that K is not a field.
 Note that the number of orbits of ρ(G_F) is equal to m and if ordered in the right way we get that |O_i|=[K_i:F].
   Choosing one orbit O:=O_i of size nd and the corresponding field E:=K_i we get a permutation representation for (E/F)≤ S_nd. 


 
	Given a transitive G≤ S_n and an integer k. We say U≤ G^k is an allowable subgroup of the natural direct product G^k if for every 1≤ i≤ k, we have π_i(U)= G where π_i: G^k → G is the natural projection to the i-th component. If O⊆{ 1, ⋯, n^k } is an orbit of U of size nd, we will say that the transitive  action of U on O is an allowable permutation subgroup H≤ S_O of G^k.








	Let F be a number field and G≤ S_n be a transitive group. Then the correctness of Conjecture <ref> for F and for all allowable permutation subgroups H of G^k  ≤ S_n^k for all k>0 implies the correctness of Conjecture <ref> for F and G.
	 



 A similar idea was used in <cit.> for F=.  Instead of counting by discriminants D they assume a Malle conjecture for counting number fields by the radical of the discriminant.
 
 We need two lemmata before we can prove this theorem.

	Given a sequence { a_m} of non-negative real numbers, then the following statements are equivalent:
	
		
  * ∀ϵ> 0: a_m =O_ϵ(m^ϵ). 
		
  * There exists an A>0 such that for all k>0: ∑_ X< m≤ 2X a_m^k = O_k(X^A).
		
  * There exists an A>0 such that for all k>0:
		∑_ m≤ X a_m^k = O_k(X^A).
	


	It is easy to see that (ii) and (iii) are equivalent. To go from (ii) to (iii) it suffices to add up over dyadic ranges. The other direction is immediate.

To go from (i) to (ii) is immediate for any A>1.   To go from (ii) to (i), for any fixed ϵ>0, we apply the following Chebychev type inequality which is easy to see here. We get for all k>0: 

    X^kϵ·∑_X<m≤ 2X, a_m> X^ϵ 1≤∑_X< m≤ 2X, a_m> X^ϵ a_m^k (ii)= O_k(X^A).

 Choosing k> A/ϵ we get that 
    ∑_X<m≤ 2X, a_m> X^ϵ 1= O_k(X^A-kϵ),
 which must be 0 when X is large enough. Therefore for any ϵ>0, there exists a X>0 such that when m> X, we have a_m ≤ m^ϵ. 


We remind the reader that (G) is defined in Definition <ref>. We denote the degree of the permutation group G by (G).

	Let G≤ S_n be a transitive permutation group, k be an integer and H be an allowable permutation subgroup of G^k. Then we get the following inequality:
		
    /(H) ≤/ (G).



Let H be an allowable permutation subgroup of G^k acting on nd points. If nd<n^k it might be that we only see 1≤ℓ≤ k projections to G.  Suppose that g∈ H is a non-trivial element. This implies that
g is a non-trivial element in at least one projection κ: H → G and
we define g̅:=κ(g). By the definition of (G) we have that (g̅)≥(G).  Let us look at all possible preimages of g̅ under κ. The index of those elements will be minimal if each cycle of length r of g̅ will decompose into d different cycles of length r. Therefore we get for a preimage g̃ of g under κ:

    #{cycles of g̃}≤ d #{cycles of g̅}.

Therefore we get: 
    (g̃) = nd - #{cycles of g̃}≥ nd - d#{cycles of g̅}
 

    = d(n-  #{cycles of g̅}) = d·(g̅).

This line is equivalent to

    1/(g̃)≤1/d·(g̅)⇔nd/(g̃)≤n/(g̅) = (G)/(g̅)≤(G)/(G).

  Note that (H)=nd and (g̃)≤(H) in order to finish the proof.

Note that in the special case H=G^k the above proof shows 

    /(G^k) = / (G)

by using (g)=(G) implies that ((g,1,…,1))=(G^k).

	We will prove that there exist constants C_1>0 and C_2>0 such that for all k>0
	
    ∑_D ≤ X a_D^k ≪_k∑_H N(F,H; C_2X^C_1 a(H))  Conj. B=O_F,ϵ,k(X^C_1+ϵ),

		where the (finite) summation goes over all allowable permutation subgroups H of G^k.
	Then the result follows from condition (iii) in Lemma <ref> by letting a_D be the number of G-extensions E/F with (E/F) = D.
	
	For proving (<ref>), it suffices to find constants C_1>0 and C_2>0 for a given allowable permutation subgroup H≤ S_nd we consider composita E:=∏_i=1^k E_i  such that (E/F) ≤ C_2X^C_1a(H) when (E_i/F) = D ≤ X for every i and (E/F) = H. Now we study the discriminant (E/F).  Recall by Hilbert's ramification theory that the valuation
	v_((E/F)) < [E:F] for an at most
	tamely ramified prime ideal ⊆ O_F. The same estimate is true for all other prime ideals of O_F lying over the same prime number p=∩. Therefore we get v_p(|(E/F)|)≤ [E:F] [F:].
	
	
	 This implies that (E/F) ≤ C_2D^[E:], where C_2 is a bounded factor coming from wildly ramified primes. Assuming D≤ X and (H)=[E:F] we get: 
    (E/F)
    	   ≤ C_2X^[E:]≤ C_2 X^[F:](H) · (/)(H)≤ C_2 X^[F:]a(H)·(/)(G)
 by Lemma <ref>. Now we see that we can take C_1 = [F:](/)(G).


	Let F be a number field.
	Assume that Conjecture <ref> is true for all solvable groups G. Then Conjecture <ref> is true for all solvable groups.
 

	Noting that all allowable (permutation) subgroups of G^k are solvable when G is solvable, the results follows directly from Theorem <ref>. 
	
In a similar way Proposition <ref> (i) can be proved for the class of solvable extensions. Here we use that subgroups of C_ℓ≀ H are solvable when H is solvable.



§ ACKNOWLEDGMENT

Wang is partially supported by Foerster-Bernstein Fellowship at Duke University, and would like to thank Melanie Matchett Wood for helpful conversations. The authors would like to thank for the hospitality of the Mathemati­sches Forschungsinstitut in Oberwolfach and the organizers of the workshop Explicit Method in Number Theory
2018 where this collaboration begins. The authors would like to thank Manjul Bhargava for many helpful conversations during the time at Oberwolfach. The authors would like to thank Brandon Alberts and Gunter Malle for suggestions on an earlier draft. This project is accomplished during the Research in Pairs (RIP) program at Mathemati­sches Forschungsinstitut in Oberwolfach in 2019, supported by the Volkswagen-Stiftung. 
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