Fundamentación:

Using the modified ray class group Cl0_m(Λ)already defined in (2.25) and Theorem 5.1.10, we can prove the following proposition.

Proposition 5.2.5 ([1, Proposition 3.9]). Let mbe an integral ideal divisible by a

sufficiently high power of |G| and of p, then we have a natural quotient

qm : Cl

0

m(Λ)−→J(KΛ)/PΘts,p

and henceλcPΘts,p is equal to a disjoint union of cosets of(KΛ)

×_·U0

Proof. We develop the proof in detail, since the original one ([1, Proposition 3.9]) contains a mistake.

If m is an integral ideal divisible by a sufficiently high power of |G| and of p

(see Theorem 5.1.10 and Proposition 2.2.9), then Θt_G(U_m0(Λ)) ⊆ Uts,p(OK[G]) and

hence Θts,p(Um0(Λ)) = 0 in Cts,p((OK[G]). Moreover Θts,p((KΛ)×) ⊆ H(K[G]).

Since every element in H(K[G]) can be obtained from a normal basis generator of a G-GaloisK-alegbra (2.1), thinking about the definition ofψts,p and of PΘts,p,

we see that (KΛ)×·U_m0(Λp)⊆ PΘts,p ⊆ J(KΛ). Then we have a natural quotient

qm : Cl0m(Λ) −→ J(KΛ)/PΘts,p, which proves our statement and shows moreover

that J(KΛ)/PΘts,p has to be finite.

From now on, we fix once and for all m to be an integral ideal satisfying the hypothesis of the previous proposition.

As we will see in the sequel the use of the modified ray class group, through the choice of the integral idealm, will oblige us to avoid some algebras in our counting procedure, but this will be clearer later on.

Let us recall some facts and notations used in [1] which are useful in the sequel. Given T a set of representatives of the orbits of G(−1) under the Ω-action, we recall the Wedderburn decomposition given in (2.23):

J(KΛ)∼= Y

t∈T

J(K(t)).

Following the description below (2.23), if we consider the set of ideals ofΛobtained taking the ideal content (denoted by co(−)) of the elements in Fp and we denote it by Fp, we see that it consists of the ideals P= (Pt)t∈T such that:

? P1 =OK,

? NKΛ/K(P) := Q

t∈T NK(t)/K(Pt) is a squarefree OK-ideal,

? Pt is coprime to the order oft,

? P is coprime top.

The function that we shall use to count extensions is defined via the use of weights introduced by Agboola in [1]. A weight on the set of representatives T, given as above, is a function W :T −→Z, which is equal to 0 fort = 1 and different from

0 for each t 6= 1. The minimum of the values it assumes outside t = 1is denoted by αW. The easiest example of weight is Wram which is defined as the constant

function 1for each t6= 1.

For every fractional ideal a= (at)t∈T of Λ, Agboola defined dW(a) :=

aW_t (t)

t∈T

and the associated discriminant for Galois algebras over K as

DW(Kh/K) := [Λ : dW(co(f))],

wheref ∈ F is the element ofJ(KΛ)satisfying (5.8) for the given extension. For example using Wram, we get that DWram(Kh/K) is equal to the absolute norm of

the product of primes of K which ramify in Kh/K.

Definition of probability. Given a class c∈ R(OK[G]) and a natural number

X, NW,m,ts,p(c, X) is defined as the number of classes in AtG(K) such that their

representativesKh/K are totally split at p, realize the classc(i.e. [OKh] =c) and

have DW(Kh/K) coprime to m and less or equal to X. While MW,m(X) denotes

the the number of classes in At_G(K) such that their representatives Kh/K have

DW(Kh/K)coprime tomand less or equal toX. In what follows we will study the

behavior of the limit of the quotient of these two quantities as X goes to infinity, which we denote as:

Pr0W,m,ts,p(c) := lim

X→∞

NW,m,ts,p(c, X)

MW,m(X)

.

We shall prove that, if W is constant on T \ {1}, this limit exists and does not depend on the class c.

Remark 5.2.6. If c /∈Rts,p(OK[G]), then Pr0W,m,ts,p(c) is clearly equal to 0.

So, given c∈ Rts,p(OK[G]), the discussion in §5.2.1 implies that

NW,m,ts,p(c, X) = K·|{f ∈ Fp∩λcPΘts,p|(co(f),m) = 1 and [Λ :dW(co(f))]≤X}|,

where K:=|Ker(ψts,p)|.

Following Proposition 5.2.5, if for every coset c of (KΛ)× ·U_m0(Λ) in J(KΛ) we define

we get NW,m,ts,p(c, X) =K ·   X c∈q−m1(λc) κW,m,ts,p(c, X)  , (5.10)

where qm is the natural quotient of Proposition 5.2.5.

Thus, we see that the asymptotic behavior of NW,m,ts,p(c, X) is controlled by the

asymptotic behavior of κW,m,ts,p(c, X). For example, if we are able to prove that

κW,m,ts,p(c, X) is asymptotically independent of c as X goes to ∞, then we will

have as a consequence that Pr0_W,m,ts,p(c) is independent of c, as we would like to prove. Note that, if κW,m,ts,p(c, X) is not asymptotically independent of c as X

goes to ∞, the asymptotic behavior of NW,m,ts,p(c, X) could still be independent

of c.