DESCRIPCIÓN DE LAN

In this section, we define the (finitely) realisable open subgroups of the Wiesend class groupCX of an arithmetical schemeX as those open subgroups whose induced

correspondence of these subgroups with open subgroups of finite index in πab 1 (X),

and provide an explicit description of these groups as the bounded open subgroups of finite index of CX, which will be proven in the following chapters. This explicit

description and its proof is the essential feature of the Main Theorem, and is the main result of this thesis.

Definition 4.3.1. An open subgroup H < CX of finite index is called finitely

realisable with realisation N if the induced covering datum is effective with finite realisation YN.

The following lemma shows that finite realisations of an induced covering datum are unique. In particular, a covering datum induced by an open subgroup of the class group has at most one finite realisation.

Lemma 4.3.2. Let X be a regular arithmetical scheme, and letN1, N2 be two open

subgroups of πab

1 (X). Then the following are equivalent:

1) N1 ⊂N2

2) ρ−_X1(N1)⊂ρ−X1(N2)

3) (ρX ◦iC∗)−1(N1)⊂(ρX ◦iC∗)−1(N2) for all curves C ⊂X

4) (ρX ◦ix∗)−1(N1)⊂(ρX ◦ix∗)−1(N2) for all closed points x∈ |X|. In particular,

an induced covering datum has at most one finite realisation.

Proof. (Taken from [5].) The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) are clear since every closed point is a regular point on some curve. If Condition 4) holds, then in

particular,N1,x⊂N2,xfor all closed pointsx, so the cover associated toN1/N1∩N2

is completely split. Hence by Lemma 2.1.12, it is trivial, which implies 1).

Proposition 4.3.3. Let X be a regular arithmetical scheme. Then the map

N 7→ ρ−_X1(N) defines a one-to-one correspondece between finitely realisable open subgroups of CX and open subgroupsN of the abelianised fundamental groupπab1 (X).

Proof. A correspondence is one-to-one if and only if it has a well-defined, two-sided inverse correspondence.

We define an inverse correspondence as follows: IfH <CX is an open subgroup of

the class group which is finitely realisable, say byfN :XN −→X, whereN / π1(X),

map H 7→ N, where N is the image of N in πab

1 (X). Then, if DH is finitely

realisable with finite realisationN, the realisation is unique by Lemma 4.3.2. Thus, the inverse correspondence is well-defined. By Proposition 4.1.5, we then also have

ρ−_X1(N) = H.

Conversely, if N is an open normal subgroup of πab₁ (X), ρ−_X1(N) is an open subgroup of CX. Let N be the preimage of N in π1(X). Let HC, Hx be the

preimages of ρ−_X1(N) in C

e

C and Cx, respectively, then by reciprocity for curves and

points, we have ρ−1 e

C (N(C)) = HC for all curves C ⊂ X, ρ

−1

x (N(x)) = Hx for all

C e C ρ e C// iC∗ π₁ab(Ce) eiC Cx ρx // ix∗ πab 1 (x) eix CX ρX// π₁ab(X) CX ρX// πab₁ (x)

In particular,HC is realised by N(C), Hx is realised by N(x) for all curves and

closed points x and we have D_ρ−1

X (N) =D

N_{. SinceN realisesD}N _{by definition, this}

shows that N realisesD_ρ−1

X (N).

We can now state the Main Theorem of this thesis. Let X be a regular arith- metical variety, and recall from that an open subgroup H ≤ CX induces a covering

datum DH.

Main Theorem 4.3.4. Let k be a finite field, and let X ⊂ _Pn

k be an open k-

subariety. Then an open subgroup H ≤ CX is finitely realisable if and only if the

induced covering datum DH geometrically bounded.

Remark 4.3.5. Lemma 4.2.13 showed that finitely realisable subgroups are geomet- rically bounded, and the converse will be shown for open affine subsets of _Pn

k in the

next chapter.

Remark 4.3.6. In the flat case of an arithmetical scheme X, the analogue of this theorem is the content of Wiesend’s paper [21], supplemented by the work of Kerz- Schmidt ([5], [6]).

Let X be an arithmetical scheme with id`ele class group CX. Recall from 3.1.9

that a subgroup is a norm subgroup iffH =f∗(N CY) for some ´etale coverY −→X.

Corollary 4.3.7. Let X ⊂_Pn

k be an open subvariety, then the following hold:

1) There exists a one-to-one correspondence between open and geometrically bounded subgroups H < CX of finite index in the Wiesend id`ele class group and open sub-

groups N of πab₁ (X); it is given byρ_X−1(N)7→N. ThenρX is a continuous injection with dense image in πab

1 (X).

2) A subgroup of finite index CX is a norm subgroup if and only if it is realisable

with a finite realisation, which is the case if and only if it is open and geometrically bounded.

3) If f : X00 −→ X is an ´etale connected cover, X0 −→ X the maximal abelian subcover, then N CX00 =N C_X0 and the reciprocity map gives rise to an isomorphism

CX/N CX00

'

−→Gal(X0/X).

Proof. We begin by proving Part 2 and note that the first equivalence is trivial. Let

H < CX be a norm subgroup of finite index, then H =N(f∗) for some ´etale cover

Y −→X. Note hat norm subgroups are open since the local norm is an open map, and the induced map f∗ :CY −→ CX was defined as the sum of local norms.

Since a norm subgroup N(f∗) is realisable the cover f it is induced from, by

Lemma 4.2.9, the norm subgroupN(f∗)<CX is geometrically bounded. In particu-

lar, a norm group of finite index is open, of finite index and geometrically bounded, as claimed. The Main Theorem 4.3.4 gives the converse statement.

Part 3 follows from 2 : Let N(f∗) =f∗(CX00) denote the f-norm subgroup, and

let DN(f∗) be the induced covering datum. Then by Part 2 of this corollary,DN(f∗)

is realisable, and the realisation fN :XN −→X is finite since it must be a subcover

of f. By Remark 4.1.5, we have

CX/N(f∗)'πab1 (X)/N 'Gal(X

0

/X)

In particular, fN is abelian.

Lastly, we show Part 1: The one-to-one correspondence follows from Proposition 4.3.3 and Part 2 of the corollary. The kernel of ρX is the connected

component of the identity (cf. [21, Section 8]), which is equal to {1} for arith- metic varieties by Remark 3.1.7. As πab

1 (X) is a profinite group, the one-to-one

correspondence implies that ρX has dense image.