Cableado Vertical

We begin by considering the affinen-spaceX =_An

k. LetA=k[x1, . . . , xn] and letD

be a geometrically bounded covering datum of cyclic index ponX =Spec A. This special case formed the starting point of the investigation, and already contains most of the strategies used for the more general cases. The goal will be to show the following Proposition:

Proposition 5.2.1. Let k be a finite field of characteristic p and let D be a bounded covering datum of index p on _An

realising D.

Throughout the section, we shall make much use of the following, particularly simple fibrations of affine space:

Definition 5.2.2. Let Ai =k[x1, . . . ,xˆi, . . . , xn] and let φi :Ai −→A,φi(xj) =xj

be the natural inclusions, then the induced morphism Φi :Ank −→A

n−1 k

are called the projection fibrations onto the coordinate hyperplanes.

Fix a projection fibration Φi. A closed pointω∈An_k−1corresponds to a maximal

ideal mω = (g1, . . . ,gˆi, . . . , gn), where gj =gj(x1, . . . ,xˆi, . . . , xj) is a polynomial in j variables ifi > j, and in j−1 variables if i≤j. Letting k(ω)'A/mω denote the

residue field ofω, the fiberCω,i of Φi aboveωis isomorphic toSpec k(ω)[xi]'A1k(ω).

As before, we let Y_CD_ω,i −→ Cω,i denote the cover induced by the subgroup NCω,i/ π1(Cω,i) of the covering datumD. Recall the notations established in Section

5.1, then by Proposition 5.1.2, there exists a polynomial fω,i ∈ k(ω)[xi] such that YCω,i = Y[fω,i] is the Artin-Schreier cover associated to [fω,i] ∈ A/℘(A). Y[fω,i] will

be referred to as an Artin-Schreier representative of the cover YCω,i.

Now consider the natural surjection

pi :Ak(ω)[xi]/℘(k(ω)[xi])

which gives a map whose kernel clearly contains ℘(A), thus inducing a surjection

For F ∈A mapping to fω,i ∈ k(ω)[xi], we thus have πi([F]) = [fω,i]. For any such F, the base change of the cover YF over Cω ,→_An _{is just Y}

[fω,i] by construction. In

particular, ifY[fω,i]is the cover induced by an index-pnormal subgroup NCω,i of the

covering datum D, then Y[F] trivialises the covering datum D over Cω.

Let F ∈ A, then we denote by deg_xi(F) the degree of F considered as a poly- nomial of k(x1, . . . ,xˆi, . . . , xn)[xi].

Lemma 5.2.3. In the notation established above, let f ∈[fω,i] be a representative

of degree d. Then there exists a class [F] ∈ π−1(f) containing an element F such that degxi(F)≤d.

Proof. Write k(ω) = k(θ1, . . . ,θˆi, θn), where θj+1 is such that

k(x1, . . . ,xi, . . . , xjˆ +1)/(g1, . . . ,gi, . . . , gjˆ +1) =k[x1, . . . ,xi, . . . , xjˆ ]/(g1, . . . ,ˆgi, . . . , gj) θj+1.

Then anya ∈k(ω) can be written asa=P

aIθI, whereI = (α1, . . . ,αi, . . . , αnˆ −1)

is a multi-index such that P

αj ≤ [k(ω) : k]. If we now define an element of A by Fa(x1, . . . ,xˆi, . . . , xn) :=

P

aIxI, then p(Fa) = a in k(ω). Moreover, given f(xi) = adxd_i +. . .+a0 ∈ k(ω)[xi], we can define F(x1, . . . , xn) = Fa_dxd_i +. . . Fa0

such that pi(F(x1, . . . , xn)) =f(xi). Since deg(f) = d, we have degxi(F) = d, and

πi([F]) =f by the remark preceding the claim.

Recall that we set A = k[x1, . . . , xn], and recall from 5.1.18 the definitions of M∆(A) and M∆i (A).

Lemma 5.2.4. LetA=k[x1, . . . , xn]and fix a positive integer∆. LetMi ⊂M∆i (A)

M ⊂ M∆(A), i.e. every equivalence class in M contains an element G such that

deg_xiG≤∆.

Proof. As M ⊂ Mi ⊂ M∆i(A) for all i, every equivalence class of M contains an

element Fi such that degxi(Fi) ≤ D. We have Fi = Fj +h

p_{−h for some h ∈ A.} Write Fi = X α aαxαi, Fj = X α0 bα0xα 0 i ,

where the coefficentsaαandbα0 are polynomials ink[x₁, . . . ,xˆ_i, . . . , x_n], and consider

the terms of Fi that are of highest xi-degree D. Note that for any non-constant h ∈ k[x1, . . . , xn], the monomials of highest xi-degree in hp−h contain all xk to a power divisible by p. Thus, if D is not divisible byp, or if aD contains a monomial

whose exponents are not all multiples ofp, andFj+hp−his equal toFi, thenhp−h

must be of xi-degree strictly smaller than D. In particular,aDxD_i is left unchanged by subtractinghp_{−h, and must thus be equal to the corresponding terms in F}

j. In

particular, we must have equality of degrees in xi: degxi(Fi) = degxi(Fj).

Now consider the case wherep divides D, and all exponents in the terms of aD

are divisible by p: Let c be the highest power of p dividing all exponents of terms in aD, and such thatpc divides D. Then we can write

aD =a0_D(xp₁c, . . . , xp_nc) =X I aIxpcα1 1 . . . x pc_α n n ,

where the sum goes over all multi-indices I = (pc_α

let a0_I be such that a0pc I =a0I, and define h= P Ia 0 IxD 0 , where D=pc_D0_{. Clearly,} hpc−h= (hpc −hpc−1) + (hpc−1 −hpc−2) +. . .+ (hp−h)∈℘(k[x1, . . . , xn]) and F_i0 =Fi+ (hp c −h)∼Fi has degxi(F 0

i)< d. Repeating the process if necessary,

we thus obtain F_i0 of xi-degree D0 such that either p does not divide D0, or such

that aD0 contains a term in which a variable occurs with an exponent not divisible

by p. We are thus reduced to the first case, and get deg_xi(Fj) = degxi(F

0

i)≤degxi(Fi)≤D .

Now use this procedure to compare F1 to all Fi, then F1 must satisfy

deg_xi(F1)≤degxi(Fi)≤D for all i, as required.

Proposition 5.2.5. Let k be a finite field, and let Φi : Ank −→ A n−1

k be the ith

coordinate fibration. Let D be a geometrically bounded covering datum of index p

on_An

k, and letY

D

Cω,i −→Cω,i be the cover ofCω,i defined byD. Then for allω andi, there exists an Artin-Schreier representative fω,i ∈k(ω)[xi] such that Y[fω,i]=Y

D

Cω,i and a positive integer ∆ such that deg_xi(fω)≤∆ .

Proof. SinceDis a covering datum of index p, for all ωandi, we have Y_CD_ω,i =Y[fω,i]

for some fω,i ∈ k(ω)[xi]. Let s = degxi(fω,i). Whenever s > 0 and p|s, the

highest term is of f is of the form axps_i 0. We may then make a change of variables

z 7→ z−a0x_is0, where a0p = a, and thus replace the term of highest degree of f by its pth root. Repeating if necessary, we may without loss of generality assume that (s, p) = 1.

Recall that Y[fω,i] denotes the normalisation of A

1 k(ω) in K(fω,i) :=k(ω)(xi)[z]/(zp −z−fω,i(xi)).

The regular compactification of _A1_k(ω) is _P1_k(ω); let Y[fω,i] denote the normalisation

of _P1

k(ω) in K(fω,i). We have a commutative diagram

Y[fω,i] // f_Cω,i Y[fω,i] f_Cω,i A1k(ω) //P 1 k(ω) ,

where all covers are defined over k(ω). The ramification locus RCω,i either consists

of the point x∞ at infinity corresponding to the fractional ideal (1/xi) ⊂k(ω)(xi),

or is empty.

Since D is assumed to be geometrically bounded, by Proposition 4.2.8, there exists a positive integer such that the ramification numbers my∞(ω, i) ≤ ∆. By

Corollary 5.1.4, we get that deg_xifω,i ≤∆ for allω, i.

Remark 5.2.6. Using Lemma 5.1.4 and the fact that the genera of degree-pcovers of affine lines can easily be computed, we can also provide a direct proof of Proposition 5.2.5 which does not make use of Proposition 4.2.8:

Indeed, the degree ofCω,i as a subvariety of Pnk is given by degkCω,i = [k(ω) :k],

and fCω,i :YCω,i −→Cω,i is defined over kCω,i =k(ω), so we have degk_Cω,iCω,i = 1

for all fibers Cω,i of one of the coordinate fibrations Φi.

P1k(ω)\A 1

k(ω), then [k(y∞) :k(x∞)] = 1. Note also that k(x∞) = k(ω), and that we

have gCω,i =gA1k(ω) = 0 for all fibers Cω,i, Cω,i\Cω,i = P

1 k(ω)\A 1 k(ω). Then Hurwitz’s formula (cf. 4.2.1) gives gY[_fω,i] = p(gCω,i −1) + 1 + 1 2degk(ω)RC = p(gCω,i −1) + 1 + 1 2 ∞ X i=0 (|Gy∞| −1) [k(y∞) :k(ω)] = 1−p+1 2 my∞ X i=0 (|Gy∞| −1) = (my∞−1)(p−1) 2 By Lemma 5.1.4, we have gY_[fω,i] = (s−1)(p−1) 2 (5.2.1)

Since D is geometrically bounded, Definition 4.2.2 gives a constant δ = δ(1) such that gY_Cω,i ≤ δ for all ω ∈ Ank−1, for all coordinate fibrations i. Thus, s is

bounded by

s≤ 2δ

p−1 + 1, which proves the Proposition as claimed.

Proof of Proposition 5.2.1. Following the notation established above, let

Y_CD

ω,i −→ Cω,i denote the induced by the covering datum D, and for each ω and i,

let fω,i ∈Aω,i be an Artin-Schreier representative of Y_CD_ω,i.

Recall that Cω,i 'A1k(ω) denotes the fiber of the ith projection fibration onto a

the fibers have degree deg_k

CCω,i = degk(ω)A

1

k(ω) = 1 over their field of definition kC =k(ω). Thus, by Proposition 5.2.5, there exists a positive integer ∆ such that deg(fω,i)≤∆ for all ω,i.

From Lemma 5.2.3, we may now find lifts Fω,i ∈ π−1([fω,i]) such that

deg_xiFω,i = deg_xifω,i ≤ ∆ for all ω, for all i = 1, . . . , n. Let Mi = ⊕ωFp.[Fω,i] be the subspace of A/℘(A) generated by the lifts {Fω,i} and recall the definition

of M∆(A) and M∆i (A) from Definition 5.1.18. Then Mi ⊂ M∆i (A) for all i and YMi −→A

n _{trivialises all NC}

ω,i by construction.

The following proposition shows that YMi is a weak trivialisation of D, i.e. YMi

trivialises Nx for all closed points x contained in a fiber Cω,i of Φi:

Proposition 5.2.7. Letf :X −→X0 be a fibration with regular fibers,Da covering datum on X. If Y −→X is a pro-´etale cover trivialising D on each fiber of f, then

Y weakly trivialises D.

Proof. Every closed point x ∈ |X| is contained in some fiber, say Cω for some ω ∈ X0. As all the fibers are regular, x is always a regular point of Cω. Then the

proposition follows directly from Lemma 2.3.18.

Now let M = hMiii, then the associated p-elementary cover YM also weakly

trivialises D by Theorem 5.0.8.

By Lemma 5.2.4, M is contained in M∆(A) = h[F] : degxiF ≤ ∆ for all ii.

generated. Thus M is of finite rank, soYM −→Ank trivialisesD by Theorem 2.3.10.

Then D is realisable with a finite realisation by Theorem 2.3.15.