In terms of this map, NIBP is reduced to the study of the kernel R(ie. In the second part of the chapter, we will adapt the uniform distribution results from [1] to Rts,p(OK[G]).
Notations
In this chapter we will recall some general results connected to the notion of locally free class group. Note that in our definition of locally free Λ-lattices of rank n R(p) (resp.
Link with projective modules
Genus and locally free class group
In order to define a group of locally free classes for a general order Λ, we need to focus on the elements in g(Λ). If Λ = R, then the locally free class group Cl(R) is exactly an ideal class group of R.
Cancellation law
The Eichler condition is in fact only a sufficient condition to have local free cancellation, indeed for example when Λ =Z[G], various authors such as J. Swan investigated groups G for which locally free cancellation holds for the integral group. bell even if the Eichler condition does not.
Other (equivalent) definitions of the locally free class group
Other equivalent definitions of the locally free class group Cl(Λ) exist, considering the following Grothendieck group. If the group Gi is abelian, then the locally free class group Cl(OK[G]) is isomorphic to the Picardy group Pic(OK[G]) (see [4, Part 4, Corollary 3.8]).
Idelic representation of Cl(Λ)
Reduced norm
As already mentioned at the beginning of the section, a reduced norm map can be defined for any field K (not necessarily a field of algebraic numbers). J(Ki)+ (subgroup of ideals positive for real primes branching into A), with reduced norm behavior for infinite real primes.
Idelic description
Locally for every finite place p, considering Ap, we have that rnAp/Z(Ap) is surjective onZ(Ap)×with kernel the commutator subgroup, whilernap/Z(Ap)(Λ×p)⊆Z(Λp ) ×, with equality whenΛp is a maximal order. There is also a way to write the previous isomorphism without considering the infinite primes, i.e.
Explicit idelic description for A = K[G]
The ideal representation can actually be derived from a more general work by Wall [44] where he gave an ideal description of Cl(Λ) with Λ an R-order, where R is now a general Dedekind domain. Note that a representative of the class of M with respect to isomorphism (1.9) is obtained by applying the reduced norm rn to the ideal element (cp)p found in the previous proposition.
Hom representation of Cl(Λ)
The Det map
To each sign χ we can associate the map detχ, an extended version of the usual determinant map, via the following diagram. At the beginning of the section we chose a Galois field expansion E/K, nevertheless, everything remains the same if we consider another field expansion E0/K, with the only requirement that it is Galois and contains all the values of the ir- reducible complex characters of G.
Hom description
We can now proceed with the proof of the statement, starting with the definition of isomorphism p. For the commutativity of the diagram, it suffices to show in a single character χ∈ Irr(G), since then by means of linearity we can extend it to RG.
Functorial properties of Cl(R[G])
Change of the group
On the rings of virtual characters we have the induced mapindGH :RH −→RG which in turn defines a mapr: Hom(RG,−)−→Hom(RH,−). The only character of {1} is the trivial χ0 and in addition G{1}(χ0) =ρG, where ρG is the character of the regular representation of G defined as ρG := P. Restricting the characters of G to H gives a constraint map on virtual characters resGH : RG −→RH, which in turn defines a map i: Hom(RH,−)−→Hom(RG,−).
Change of the base field
G-Galois K-algebras
Henceforth m will denote the exponent of G (i.e. the least common multiple of the order of the elements of G). Then we give the unpublished proof of the following result (this corresponds to Theorem 0.0.5 in the Introduction). If we denote by H1(Ωnrp, G) the elements in H1(Ωp, G) that come from the homomorphisms hinHom(Ωnrp, G), thanks to the local version of the previous proposition, we have a local integral analog of the exact sequence (2.3. ).
Determinants of resolvends
Going back to the previous short exact sequence and using the fact that (Kc)× is injective, the function Hom(−,(Kc)×) yields the next short exact sequence. If we extend the definition of the mapDet in section 1.7.1 to Det :Kc[G]×−→Hom RG,(Kc)× we now have the following statement. Turning now to Rnr(OK[G]), the subset of realizable classes given by unbranched (in finite places)G-GaloisK algebras, using the previous results on resolvends and via §1.6.11, we have the following result.
The Stickelberger map
If we let Ω act trivially on G, as at the beginning of the chapter, it is easy to see that the Stickelberger map does not preserve the action Ω. To have such an invariant property, we need to introduce a non-trivial action Ω on G. If we denote by G¯ the set of conjugacy classes of G, then the action of Ωviaκ−1 preserves the conjugacy classes and causes an action Ω . in Z(Z[G]) and in Z[ ¯G]; we denote these Ω-modules by Z(Z[G(−1)]) and Z[ ¯G(−1)], respectively.
Resolvends of local tame extensions
Here we must make a clarification: the theory of resolution values that we have seen in the previous sections for G-Galois algebras over K (or Kp) can be generalized to the set of Hopf algebras, as fully done by N. The fact, that rG(a)∈ Kpc[G]× proves that in particular it is a normal basis generator of (Kp)hp/Kp. The equality follows, as in the Abelian proof, by comparing the discriminants of OK,p[G]·and of O(Kp)h.
Proof of Theorem A
Then, as in Proposition 1.6.12, we choose a normal basis generator b and for each finite placep a local normal integral basis generatorap.
The abelian equality
Then, by the previous proposition, given an element g ∈ J(KΛ) there exists an f ∈ F (which can be disjointly disjointly resolved from any predetermined finite set of finite primes S), i such that Furthermore, RF(OK[G]) =RA(OK[G]) and any class can be obtained from an extension of the unbranched G-Galois Kh/K field to a finite predefined set of primes of the end. The statement on branching follows from the fact that the element f ∈ F in a given set of the class of modified rays can be chosen with disjoint support from any given predetermined set of primes.
Computing St(O K [G])
- Background on C p and D p
- Stickelberger’s classical theorem
- The Stickelberger map for C p and D p
- The augmentation kernels A C p and A D p
- The triviality of Θ t C
- Proof of Theorem 3.2.1
While for the two-level group Dp (withp≥3) we must first think about the restriction of the irreducible characters over the cyclic subgroupshri(of orderp) andhsrki(of order 2). So we can now calculate the transpose of the Stickelberger map (again when p= 2 we consider only the first two lines) onh∈ HomΩ(Z[Cp(−1)], J(Q(ζp))), obtain. For the second one, it is sufficient to keep in mind the definition of the Stickelberger subgroup and use Proposition 4.1.1.
Corollaries
Resolvends of local totally split extensions
The previous equality could also be deduced from the exact sequence (2.5), namely the set of reduced resolutions of NIBG for totally split G-Galois algebras over Kp is the core of the connecting homomorphism H(OK,p[G]) −→ H1(Ωnrp , G). By the previous comment, considering the exact sequence (2.5), if we show that H1(Ωnrp , G) is finite, we have the claim. Another way to prove the previous lemma, which is only valid in the abelian case, is as follows: since G is finite and abelian, it has a finite exponent denoted by m, so that is easy to see.
The subgroup R ts,p (O K [G])
Furthermore, any class can be obtained from an extension of the unbranched G-Galois field to a predefined finite set of finite primes of K. Then if Stts,p(OK[G]) is defined the same as the situation abelian (5.7), an analogue of theorem A follows. Let G be a finite group and let K be a number field, then for every finite place p of K, we have.
Proof of Theorem 5.1.10
If Rts,S(OK[G]) denotes the set of feasible classes given by tame G-Galois K-algebras that are completely split at every prime inS, we can generalize the proof of Theorem 5.1.11 and obtain Rts,S(OK [ G]) = Stts,S(OK[G])(with G abelian of course!). Now given M and H as before, for each n ∈ N we can consider a continuous H-equivariant automorphismn:M →M which sendstomn (it is continuous because M is a topological group and multiplication is continuous by definition). Using these lemmas, we can now proceed with the proof of Theorem 5.1.10, but before doing so, we make the following remark.
Equidistribution for R ts,p (O K [G])
Towards the counting problem
The independence of ψts,p from the choice of b follows from the fact that we quotient by K[G]×. In addition to counting all the tamG-GaloisK algebras (up to isomorphisms) totally split at p and realizing a fixed class [c] ∈ Rts,p(OK[G]), we must count all the pairs (h, f ), meth∈ Hom(Ω, G) andf ∈ Fp, satisfying equation (5.8), where [c] is fixed. Therefore we see that the counting of the pairs(h, f), withh∈ Hom(Ωt, G) and f ∈ Fp, satisfies equation (5.8), with[c] fixed (and therefore all the tame G-Galois K count -algebras, up to isomorphisms, totally split upon prerealization of the class[c]), is the same (up to multiplication by|Ker(ψts,p)|) of the count of the elements inFp∩λcPΘts,p for a fixed compositionλcPΘts , p of PΘts,p inJ(KΛ).
Algebraic part
From now on, we establish once and for all that it is an integral ideal that satisfies the hypothesis of the previous proposition. As we shall see below, the use of the modified ray class group, through the choice of the integral ideal, will oblige us to avoid some algebras in our counting procedure, but this will become clearer later. For example, using Wram, we get that DWram(Kh/K) is equal to the absolute norm of the product of primes of K that branch into Kh/K.
Analytic part
Since in our case we consider divisible by a sufficiently high power of|G|and of p, we understand that, by the description of Fp given above, we have. We therefore reduced our problem to study the asymptotic behavior of the sum of the first X coefficients of a convergent Dirichlet series. The analogue of [1, Theorem B] therefore now clearly follows (this is a more precise version of Theorem 0.0.9 in the Introduction).
Probabilities on G-Galois K-algebras totally split at p
More primes and other types of splitting behavior
Le théorème des bases normales nous dit qu'une algèbre de Galois N/K est un module libre K[G] de rang 1, c'est-à-dire Explicitement, l'ensemble des classes possibles est l'ensemble des classes de Cl(OK[G]) que l'on peut obtenir à partir des anneaux d'algèbres de Galois entiers sur K, du groupe de Galois G, modérément ramifiés. Tout d'abord, nous avons décrit, sous la forme d'une version modifiée de St(OK[G]), l'ensemble des classes de possibles obtenues à partir de développements de Galois moyennement ramifiés complètement décomposés en un idéal premier donné p⊆OK.
