2 Semi-graphs with p-rank

Local p-Rank and Semi-Stable Reduction of Curves

Yu Yang

Abstract

In the present paper, we investigate the localp-ranks of coverings of stable curves.

LetGbe a finitep-group,f :Y −→X a morphism of stable curves over a complete discrete valuation ring with algebraically closed residue field of characteristicp >0, xa singular point of the special fiberX_s ofX. Suppose that the generic fiberX_η of X is smooth, and the morphism of generic fibers f_η is a Galois ´etale covering with Galois group G. Write Y^′ for the normalization of X in the function field of Y, ψ:Y^′ −→Xfor the resulting normalization morphism. Let y^′ ∈ψ⁻¹(x) be a point of the inverse image ofx. Suppose that the inertia groupI_y′ ⊆Gofy^′ is an abelian p-group. Then we give an explicit formula for thep-rank of a connected component of f⁻¹(x). Furthermore, we prove that the p-rank is bounded by ♯I_y′ −1 under certain assumptions, where ♯I_y′ denotes the order of I_y′. These results generalize the results of M. Sa¨ıdi concerning local p-ranks of coverings of curves to the case whereI_y′ is an arbitrary abelian p-group.

Keywords: p-rank, semi-stable reduction, semi-stable covering, semi-graph with p-rank.

Mathematics Subject Classification: Primary 14E20; Secondary 14H30.

Contents

1 Introduction and ideas 2

2 Semi-graphs with p-rank 4

2.1 Definitions . . . 5 2.2 p-ranks and ´etale-chains of abelian coverings . . . 8 2.3 Bounds of p-ranks of abelian coverings . . . . 18 3 p-ranks of vertical fibers of abelian stable coverings 23 3.1 p-ranks and stable coverings . . . . 23 3.2 Semi-graphs withp-rank associated to vertical fibers . . . . 27 3.3 p-ranks of vertical fibers . . . . 30

4 Appendix 32

1 Introduction and ideas

LetRbe a complete valuation ring with algebraically closed residue fieldkof characteristic p >0, K the quotient field of R, and K an algebraic closure of K. We use the notation S to denote the spectrum of R. Write η, η and s for the generic point, the geometric generic point, and the closed point corresponding to the natural morphisms SpecK −→S, SpecK −→S, and Speck −→S, respectively. Let X be a stable curve of genus g_X over S. WriteXη,Xη, and Xs for the generic fiber, the geometric generic fiber, and the special fiber, respectively. Moreover, we suppose that X_η is smooth over η.

Let Y_η be a geometrically connected curve overη, f_η :Y_η −→X_η a finite Galois ´etale covering over η with Galois group G. By replacing S by a finite extension of S, we may assume that Y_η admits a stable model over S. Then f_η extends uniquely to a G-stable covering(cf. Definition 3.3)f :Y −→X overS(cf. [L2, Theorem 0.2] or Remark 3.3.1 of the present paper). We are interested in understanding the structure of the special fiber Y_s ofY. If the order♯GofGis prime top, then by the specialization theorem for log ´etale fundamental groups, f_s is an admissible covering (cf. [Y1]); thus, Y_s may be obtained by gluing together tame coverings of the irreducible components of Xs. On the other hand, if p|♯G, then f_s is not a finite morphism in general. For example, if char(K) = 0 and char(k) = p >0, then there exists a Zariski dense subset Z of the set of closed points of X, which may in fact be taken to be X when k is an algebraic closure of Fp, such that for any x ∈ Z, after possibly replacing K by a finite extension of K, there exist a finite group H and an H-stable covering f_W : W −→ X such that the fiber (f_W)⁻¹(x) is not finite (cf. [T], [Y2]).

Iff⁻¹(x) is not finite, we shall callxavertical point associated tof and callf⁻¹(x) the vertical fiber associated to x (cf. Definition 3.4). In order to investigate the properties of Ys, we focus on a geometric invariantσ(Ys) which is called thep-rank ofYs (cf. Definition 3.1 and Remark 3.1.1). By the definition of the p-rank of a stable curve, to calculate σ(Y_s), it suffices to calculate the rank of H¹(Γ_Ys,Z) (where Γ_Ys denotes the dual graph of Ys), the p-ranks of the irreducible components of Ys which are finite over Xs, and the p-ranks of the vertical fibers of f. In the present paper, we study the p-rank of a vertical fiber and consider the following problem:

Problem 1.1. Let G be a finite p-group, x be a vertical point associated to the G-stable covering f :Y −→X, f⁻¹(x) the vertical fiber associated to x.

(a) Does there exist a minimal bound on the p-rank σ(f⁻¹(x)) (note that σ(f⁻¹(x)) is always bounded by the genus of Ys)?

(b) Does there exist an explicit formula for the p-rank σ(f⁻¹(x))?

We will answer Problem 1.1 under certain assumptions (cf. Theorem 1.5 and Theorem 1.10). First, let us review some well-known results concerning Problem 1.1.

Ifx is a nonsingular point, M. Raynaud proved the following result (cf. [R, Th´eor`eme 1]):

Theorem 1.2. If x is a non-singular point of X_s, and G is an arbitrary p-group, then the p-rank σ(f⁻¹(x)) is equal to 0.

By Theorem 1.2, in order to resolve Problem 1.1, it is sufficient to consider the case where x is asingular point of X_s. In order to explain the results obtained in the present paper, let us introduce some notations. Write X₁ and X₂ for the irreducible components ofX_s which containx,ψ :Y^′ −→X for the normalization of X in the function field ofY. Lety^′ ∈ψ⁻¹(x) be a point in the inverse image ofx. Write I_y′ ⊆G for the inertia group of y^′. In order to calculate thep-rank of f⁻¹(x), since Y /I_y′ −→X is finite ´etale overx, by replacingX by the stable model of the quotientY /I_y′ (note thatY /I_y′ is a semi-stable curve overS (cf. [R, Appendice, Corollaire])), we may assume that G is equal toI_y′.

Thus, from the point of view of resolving Problem 1.1, we may assume without loss of generality thatG=I_y′. In the remainder of this section, we shall assume that G=I_y′

is of order p^r for some positive integer r. Then f⁻¹(x) is connected. With regard to Problem 1.1 (a), M. Sa¨ıdi proved the following result (cf. [S, Theorem 1]), by applying Theorem 1.2:

Theorem 1.3. If G is a cyclic p-group, then we have σ(f⁻¹(x)) ≤ ♯G−1, where ♯G denotes the order of G.

Furthermore, there is an open problem posed by Sa¨ıdi as follows (cf. [S, Question]):

Problem 1.4. If G is an arbitrary p-group, does there exist a minimal bound on the p-rank σ(f⁻¹(x)) that depends only on the order♯G?

Let us introduce some notations. Suppose that G is an abelian p-group. Let Φ :{1}=G_r⊂G_r−1 ⊂ · · · ⊂G₀ =G

be a maximal filtration of G (i.e., G_i/G_i+1 ∼= Z/pZ for i = 0, . . . , r−1). It follows from [R, Appendice, Corollaire], that fori= 0, . . . , r, Y_i :=Y /G_i is a semi-stable curve overS.

Write X^sst forY /G and g for the resulting morphism g :X^sst−→X induced byf. Then we obtain a sequence of Z/pZ-semi-stable coverings (cf. Definition 3.3)

Φ_f :Y =Y_r −−−→^dr Y_r−1 −−−→^dr−1 . . . −−−→^d1 Y₀ =X^sst −−−→^g X.

In the following, we use the subscript “red” to denote the reduced induced closed sub- scheme associated to a scheme. For eachi= 1, . . . , r, writeϕ_i :Y_i −→Y₀for the composite morphismd₁◦· · ·◦d_i. For simplicity, we suppose thatC :=g⁻¹(x)_red =∪ⁿj=1P_j, where, for each j = 1, . . . , n, P_j is isomorphic to P¹ and meets the other irreducible components of the special fiberX_s^sst of X^sst at precisely two points (i.e., a chain of P¹). Thus, thep-rank σ(f⁻¹(x)) is equal to σ(ϕ⁻_r¹(C)). For each i= 1, . . . , r, we define a set of subcurves of C associated to Φ_f, which plays a key role in the present paper, as follows: ♦

Ei^Φf :=ϕ_i(the ´etale locus of d_i|_ϕ⁻¹

i (C)red :ϕ⁻_i ¹(C)_red −→ϕ⁻_i−¹₁(C)_red)⊂C.

We shall callEi^Φf the i-th ´etale-chain associated to Φ_f and call the disjoint union E^Φf :=⨿

i

Ei^Φf

the ´etale-chain associated to Φ_f. For each connected component E of E_i^Φf, we use the notation l(E) to denote the cardinality of the set of the irreducible components of E and call l(E) the length ofE.

We generalize Sa¨ıdi’s result as follows (see also Theorem 3.15):

Theorem 1.5. If G is an arbitrary abelian p-group, and Ei is connected for each i = 1, . . . , n, then we have σ(f⁻¹(x))≤♯G−1.

Remark 1.5.1. If ♯G is equal to p, then we may construct a Z/pZ-stable covering f : Y −→Xsuch that there exists a singular vertical pointxsuch that thep-rank ofσ(f⁻¹(x)) is equal to p−1 (cf. [Y4, Section 4]). Thus, at least in the case where ♯G=p, ♯G−1 is the minimal bound forσ(f⁻¹(x)).

Next, let us consider Problem 1.1 (b). Let {V_i}ⁿ⁺¹_i=0 be a set of irreducible components of the special fiberY_s ofY such that the following conditions are satisfied: (i)ϕ_r(V_i) = P_i if i= 1, . . . , n; (ii) ϕ_r(V₀) =X₁ and ϕ_r(V_n+1) =X₂; (iii) the union ∪ⁿ⁺¹i=0V_i is a connected semi-stable subcurve of the special fiber Y_s of Y. Write I_Pi ⊆G for the inertia subgroup of V_i. Note that since G is an abelian p-group,I_Pi does not depend on the choices ofV_i.

If Gis a cyclic p-group, Sa¨ıdi obtained an explicit formula of thep-rank σ(f⁻¹(x)) as follows (cf. [S, Proposition 1]):

Theorem 1.6. If G is a cyclic p-group, and I_P0 is equal to G, then we have σ(f⁻¹(x)) =♯(G/I_min)−♯(G/I_Pn+1),

where I_min denotes the group ∩ⁿ⁺¹i=0I_Pi.

For aG-covering of semi-graphs withp-rank, we develop a general method to compute thep-rank (cf. Theorem 2.8). As an application, we generalize Sa¨ıdi’s formula to the case where Gis an arbitrary abelian p-group as follows (cf. Theorem 3.9 and Remark 3.9.1):

Theorem 1.7. If G is an arbitrary abelian p-group, then we have

σ(f⁻¹(x)) =

∑n

i=1

♯(G/I_Pi)−

∑n+1

i=1

♯(G/(I_Pi−1 +I_Pi)) + 1.

Finally, I would mention that by using the theory of semi-graphs with p-rank, we can generalize Theorem 1.8 to the case whereGis an arbitrary p-group. Furthermore, we can obtain a global p-rank formula for the special fiberYs (cf. [Y5]).

The present paper contains two parts. In Section 2, we develop the theory of semi- graphs withp-rank and calculate the p-ranks of G-coverings. In Section 3, we construct a semi-graph with p-rank from a vertical fiber of a G-stable covering in a natural way and apply the results of Section 2 to prove Theorem 1.5 and Theorem 1.8.

2 Semi-graphs with p-rank

In this section, we develop the theory of semi-graphs withp-rank. We always assume that G is an abelian p-group with orderp^r.

2.1 Definitions

We begin with some general remarks concerning semi-graphs (cf. [M]). A semi-graph G consists of the following data: (i) A set VG whose elements we refer to as vertices; (ii) A set E^G whose elements we refer to as edges. Any element e∈ E^G is a set of cardinality 2 satisfying the following property: For any e̸=e^′ ∈ E^G, we have e∩e^′ = Ø; (iii) A set of maps {ζ_e^G}e∈E^G such that ζ_e :e −→ V ∪ {V} is a map from the set e to the set V ∪ {V}. For an edge e ∈ E^G, we shall refer to an element b ∈ e as a branch of the edge e. An edge e ∈ E^G is called closed (resp. open) if ζ_e⁻¹({V^G}) = Ø (resp. ζ_e⁻¹({V^G}) ̸= Ø). A semi-graph will be called finite if both its set of vertices and its set of edges are finite.

In the present paper, we only consider finite semi-graphs. Since a semi-graph can be regarded as a topological space, we shall callG a connected semi-graph ifG is connected as a topological space.

LetGbe a semi-graph. Writev(G) for the set of vertices ofG,e(G) for the set of closed edges of G, and e^′(G) for the set of open edges of G. For any element v ∈ v(G), write b(v) for the set of branches ∪e∈e(G)∪e^′(G)ζ_e⁻¹(v). For any element e∈ e(G)∪e^′(G)), write v(e)for the set which consists of the elements ofv(G) which are abutted bye. A morphism between semi-graphs G −→ H is a collection of maps v(G) −→ v(H); e(G)∪e^′(G) −→

e(H)∪e^′(H); and for each e_G ∈ e(G)∪e^′(G) mapping to e_H ∈ e(H)∪e^′(H), a bijection e_G →^∼ e_H; all of which are compatible with the {ζ_e^G}e∈e(G)∪e^′(G) and {ζ_e^H}e∈e(H)∪e^′(H).

A sub-semi-graph G^′ ofGis a semi-graph satisfying the following properties: (i)v(G^′) (resp. e(G^′)∪e^′(G^′)) is a subset of v(G) (resp. e(G)∪e^′(G)); (ii) If e ∈e(G^′), then we have ζ_e^G′(e) = ζ_e^G(e); (iii) If e ={b₁, b₂} is an element of e^′(G^′) such that ζ_e^G(b₁) ∈v(G^′) and ζ_e^G(b₂)̸∈v(G^′), then we have ζ_e^G′(b₁) =ζ_e^G(b₁) and ζ_e^G′(b₂) ={v(G^′)}.

Definition 2.1. Let G^′ be a sub-semi-graph of a semi-graph G. We define a semi-graph G\G^′ as follows: (i) The set of verticesv(G\G^′) isv(G)\v(G^′); (ii) The set of closed edges e(G\G^′) ise(G)\e(G^′); (iii) The set of open edgese^′(G\G^′) is{e∈e(G)|v(e)∩v(G\G^′)̸= Ø in G}; (iv) For any e={b_i}i={1,2} ∈e(G\G^′)∪e^′(G\G^′), we have ζe^G\G′(b_i) =ζ_e^G(b_i) (resp. ζe^G\G′(b_i) ={v(G\G^′)}) if ζ_e^G(b_i)̸∈v(G^′) (resp. ζ_e^G(b_i)∈v(G^′)).

Definition 2.2. (a) Let n be a positive natural number and Pn a semi-graph such that the following conditions hold: (i) v(Pn) = {p₁, . . . , p_n}, e(Pn) = {e_1,2, . . . , e_n,n−1} and e^′(Pn) ={e0,1, en,n+1}; (ii) v(ei,i+1) ={pi, pi+1}; (iii) v(e0,1) ={p1} and v(en,n+1) ={pn}. We define G to be a triple (G, σ_G, β_G) which consists of a semi-graph G, a map σ_G : v(G) −→ Z and a morphism of semi-graphs β_G : G −→ Pn. We shall call G a n-semi- graph withp-rank. We shall refer toGas the underlying semi-graph ofG,σ_G as thep-rank map ofG,β_G as the base morphism ofG, respectively. We defineP_n := (Pn, σ_Pn, β_Pn) as follows: σ_Pn(p_i) is equal to 0 for eachi= 1, . . . , n, andβ_Pn = id_Pn is an identity morphism of semi-graph Pn. We shall call Pn an-chain.

(b) We define the p-rank σ(G) of G as follows:

σ(G) := ∑

v∈v(G)

σ(v) + ∑

Gi∈π0(G)

rank_ZH¹(Gi,Z), where π₀(−) denotes the set of connected components of (−).

(c) G is called connected if the underlying semi-graphG is a connected semi-graph.

From now on, we only consider connected n-semi-graphs with p-rank. Let G¹ :=

(G¹, σ_G¹, β_G¹) and G² := (G², σ_G², β_G²) be two n-semi-graphs with p-rank. A morphism betweenG¹andG² is defined by a morphism of the underlying semi-graphsβ :G¹ −→G² such that β_G² ◦β = β_G¹. We use the notation b : G¹ −→ G² to denotes the morphism of semi-graphs with p-rank determined by β : G¹ −→ G² and call β the underlying morphism of b. Note that for any n-semi-graph with p-rank G := (G, σ_G, β_G), there is a natural morphismb_G :G−→P_n determined by the morphism of underlying semi-graphs β_G :G−→Pn.

Write bⁱ_l (resp. bⁱ_r) forζ_e⁻_i−1,i¹ (p_i) (resp. ζ_e⁻¹

i,i+1(p_i)). For any element v_i ∈β_G⁻¹(p_i), write b_l(v_i) (resp. b_r(v_i)) for the set

{b ∈b(v_i) | β_G(b) = bⁱ_l} (resp. {b∈b(v_i) | β_G(b) =bⁱ_r}).

Definition 2.3. Letb : G¹ := (G¹, σ_G¹, β_G¹)−→G² := (G², σ_G², β_G²) be a morphism of n-semi-graphs with p-rank, β the underlying morphism of b,e ∈e(G¹)∪e^′(G¹) an edge, v₁ a vertex of G¹ contained in β_G⁻1¹(p_i), andv₂ :=β(v₁)∈β_G⁻2¹(p_i) the image of v₁.

(a) We shall call b p-´etale (resp. p-purely inseparable) at e if ♯β⁻¹(β(e)) = p (resp.

♯β⁻¹(β(e)) = 1). We shall call b p-generically ´etale at v₁ ∈β_G⁻1¹(p_i) if one of the following

´

etale types holds:

(Type-I) ♯β⁻¹(v2) =p and σ_G¹(v1) =σ_G²(v2);

(Type-II) ♯β⁻¹(v₂) = 1, ♯b_l(v₁) =p♯b_l(v₂), ♯b_r(v₁) =p♯b_r(v₂), and σ_G¹(v₁)−1 =p(σ_G²(v₂)−1);

(Type-III) If ♯β⁻¹(v₂) = 1, ♯b_l(v₁) = ♯b_l(v₂), ♯b_r(v₁) = p♯b_r(v₂), and σ_G¹(v₁)−1 =p(σ_G²(v₂)−1) + (♯b_l(v₁))(p−1);

(Type-IV) ♯β⁻¹(v₂) = 1, ♯b_l(v₁) = p♯b_l(v₂),♯b_r(v₁) =♯b_r(v₂), and σ_G¹(v₁)−1 = p(σ_G²(v₂)−1) + (♯b_r(v₁))(p−1);

(Type-V) ♯β⁻¹(v₂) = 1,♯b_l(v₁) =♯b_l(v₂),♯b_r(v₁) =♯b_r(v₂), and σ_G¹(v1)−1 =p(σ_G²(v2)−1) + (♯bl(v1) +♯br(v1))(p−1).

(b) We shall call bpurely inseparableatv₁ ∈β_G⁻1¹(p_i) if♯β⁻¹(v₂) = 1,♯b_l(v₁) =♯b_l(v₂),

♯b_r(v₁) =♯b_r(v₂), andσ_G¹(v₁) =σ_G²(v₂) hold.

(c) We shall call b a p-covering if the following conditions hold: (i) There exists a Z/pZ-action (which may be trivial) on G¹ (resp. a trivial Z/pZ-action on G²), and the underlying morphism β of b is compatible with the Z/pZ-actions. Then the natural morphism G¹/Z/pZ−→G² induced by b is an isomorphism; (ii) For any v ∈v(G¹),b is eitherp-generically ´etale or purely inseparable at v; (iii) Lete∈e(G¹) and v(e) ={v, v^′}. If b is p-generically ´etale at v and v^′, then b is p-´etale at e; (iv) For any v ∈v(G¹), then σ_G¹(v) = σ_G¹(τ(v)) holds for eachτ ∈Z/pZ.

Note that by the definition of p-covering, the identity morphism of a semi-graph with p-rank is ap-covering.

(d) We shall call b a covering if b is a composite of p-coverings.

(e) We shall call

Φ :{1}=G_r ⊂G_r−1 ⊂ · · · ⊂G₁ ⊂G₀ =G

an maximal filtration of G if G_j/G_j+1 ∼= Z/pZ for each j = 1, . . . , r−1. Suppose that G¹ (resp. G²) admits a (resp. trivial) G-action (which may be trivial). Then for any maximal filtration Φ ofG, there is a sequence of semi-graphs induced by Φ:

G¹ =Gr βr

−−−→ Gr−1 βr−1

−−−→ . . . −−−→^β1 G0,

whereGj denotes the quotient ofG¹ byG_j. We shall callbaG-coveringif for any maximal filtration Φ of G, there exists a set of p-coverings {bj : Gj −→ Gj−1, j = 1, . . . , r} such that the following conditions hold: (i) the underlying morphismβ of bis compatible with the G-actions, and the natural morphism G¹/G−→G² induced byβ is an isomorphism;

(ii) The underlying graph of Gj is equal to Gj for each j = 0, . . . , r; (iii) The underlying morphism Gj −→ Gj−1 of b_j is equal to β_j for each j = 1, . . . , r; (iv) The composite morphism b₁◦ · · · ◦b_r is equal tob. Then we obtain a sequence ofp-coverings:

Φ_G¹ :G¹ =G_r −−−→^br G_r−1 −−−→^br−1 . . . −−−→^b1 G₀ =G². We shall call Φ_G¹ a sequence of p-coverings induced by Φ.

(f) LetGbe an-semi-graph withp-rank. We shall callGacovering(resp. G-covering) over P_n if b_G is a covering (resp. G-covering).

(g) Let b : G¹ −→ G² be a G-covering, v ∈ v(G) a vertex, and e ∈ e(G)∪e^′(G) an edge. For any subgroup H ⊆ G, by Definition 2.3 (e), there exists a maximal filtration Φ^H and the sequence of p-coverings

Φ^H_G1 :G¹ =G_r ^b

Hr

−−−→ G_r−1 ^b

Hr−1

−−−→ . . . ^b

H

−−−→1 G₀ =G²

induced by Φ^H such that there existsisuch that the underlying graph ofGi is isomorphic toG¹/H. We writeG¹/H forG_i. Thus, the natural morphismb^H₁ ◦· · ·◦b^H_i :G¹/H −→G² is a covering. Then we define five subgroups of Gas follows:

D_v :={τ ∈G |τ(v) =v},

I_v := the maximal element of {H ⊆G| G¹ −→G¹/H is purely inseparable at v}, I_v^l(b) :={τ ∈D_v |τ(b) = b for a branch b∈b_l(v)}/I_v,

I_v^r(b) := {τ ∈D_v | τ(b) =b for a branchb∈b_r(v)}/I_v, I_e :={τ ∈G | τ(e) = e}.

We shall callD_v (resp. I_v, I_v^l(b),I_v^r(b),I_e) the decomposition group ofv (resp. the inertia group of v, the inertia group of a left branch b, the inertia group of a right branch b, the inertia group ofe). Moreover, sinceGis an abelianp-group, the groupI_v^l(b) (resp. I_v^r(b))

does not depend on the choice of b ∈ b_l(v) (resp. b ∈ b_r(v)), then we denote this group briefly by I_v^l (resp. I_v^r). Define

D_v^e =D_v/(I_v^l/(I_v^l ∩I_v^r)⊕I_v^r/(I_v^l ∩I_v^r)⊕I_v^l ∩I_v^r⊕I_v).

Then we have the following exact sequence

0−→I_v^l/(I_v^l ∩I_v^r)⊕I_v^r/(I_v^l ∩I_v^r)⊕I_v^l ∩I_v^r⊕I_v −→D_v −→D_v^e−→0.

Remark 2.3.1. LetGbe aG-covering overP_nandv_i ∈β_G⁻¹(p_i) a vertex of the underlying graph ofG. Then we have the following Deuring-Shafarevich type formula (cf. Proposition 3.2 for the Deuring-Shafarevich formula for curves)

σ_G(v_i)−1 =−♯D_vi/I_vi+♯((D_vi/I_vi)/I_v^l_i)(♯I_v^l_i −1) +♯((D_vi/I_vi)/I_v^r_i)(♯I_v^r_i −1).

Let G be a G-covering over Pn. By the definition of G-coverings, for any maximal filtration Φ of G, we have a sequence ofp-coverings ofn-semi-graphs withp-rank

Φ_G :G=G_r −−−→^br G_r−1 −−−→^br−1 . . . −−−→^b1 G₀ =P_n induced by Φ. For each j = 1, . . . , r, we write Vj^´et for the set

{v ∈v(Gj)| b_j is ´etale atv}, Ej^´et for the set

{e∈e(Gj)∪e^′(Gj)| b_j is ´etale at e}.

Since (Vj^´et,Ej^´et) admits a natural structure of semi-graph induced by Gj, we may regard (V_j^´et,E_j^´et) as a sub-semi-graph of Gj. Thus, the image β_Gj((V_j^´et,E_j^´et)) can be regarded as a sub-semi-graph of Pn.

Definition 2.4. We shall call E^Φj^G := β_Gj((Vj^´et,Ej^´et)) (resp. the disjoint union E^ΦG :=

⨿

jE^Φj^G) the j-th ´etale-chain (resp. the´etale-chain) associated to Φ_G.

2.2 p-ranks and ´ etale-chains of abelian coverings

LetG := (G, σ_G, β_G) be a G-covering over Pn. We introduce two operators for G.

Operator I: First, let us define a G-covering G^∗[p_i] over P_n. For any p_i ∈ v(Pn), let v_i be an element of β_G⁻¹(p_i).

If ♯β_G⁻¹(p_i) = 1 (i.e., D_vi = G), then we define G^∗[p_i] to be G; If ♯β_G⁻¹(p_i) ̸= 1, we define a new semi-graph G^∗[p_i] as follows.

Definev(G^∗[p_i]) (resp. e(G^∗[p_i])∪e^′(G^∗[p_i])) to be the disjoint union (v(G)\β_G⁻¹(p_i))⨿ {v^∗} (resp. e(G)∪e^′(G)).

The collection of maps {ζe^G∗[pi]}e is as follows: (i) For any branch b ̸∈ ∪_v∈β⁻¹

G (pi)b(v), ζe^G∗[pi](b) = ζ_e^G(b) if b ∈ e and ζe^G∗[pi](b) = Ø if b ̸∈ e; (ii) For any v ∈ β_G⁻¹(p_i) and any branch b∈b(v),ζe^G∗[pi](b) = v^∗ if b ∈e and ζe^G∗[pi](b) = Ø if b̸∈e.

We define a map σ_G∗[pi]:v(G^∗[p_i])−→Z as follows: (i) Ifv^∗ ̸=v ∈v(G^∗[p_i]), then we haveσ_G∗[pi](v) := σ_G(v); (ii) Ifv =v^∗, then we have

σ_G∗[pi](v^∗) :=−♯(G/I_vi) + ∑

v∈β_G⁻¹(pi)

∑

b∈b_l(v)

(♯I_v^l(b)−1) + ∑

v∈β_G⁻¹(pi)

∑

b∈br(v)

(♯I_v^r(b)−1) + 1

=−♯(G/Ivi) +♯((G/Ivi)/I_v^l_i)(♯I_v^l_i −1) +♯((G/Ivi)/I_v^r_i)(♯I_v^r_i−1) + 1.

We define a morphism of semi-graphs β_G∗[pi] : G^∗[p_i] −→ Pn as follows: (i) For any v ∈ v(G^∗[p_i]), β_G∗[pi](v) = p_i if v = v^∗ and β_G∗[pi](v) = β_G(v) if v ̸∈ β_G⁻¹(p_i); (ii) If e∈e(G^∗[p_i])∪e^′(G^∗[p_i]), then we have β_G∗[pi](e) =β_G(e).

Thus, the triple G^∗[p_i] := (G^∗[p_i], σ_G∗[pi], β_G∗[pi]) is a n-semi-graph with p-rank.

Moreover,G^∗[p_i] admits a naturalG-action as follows: (i) the action ofGonv(G^∗[p_i])\ {v^∗}(resp. e(G^∗[p_i])∪e^′(G^∗[p_i])) is the action ofGonv(G)\β_G⁻¹(p_i) (resp. e(G)∪e^′(G));

(ii) For any τ ∈G, we have τ(v^∗) =v^∗.

Let us explain that with the G-action defined above, G^∗[p_i] is a G-covering over P_n. Let

Φ :{1}=G_r ⊂G_r−1 ⊂ · · · ⊂G₁ ⊂G₀ =G be an arbitrary maximal filtration of G. Write

Φ_G :G=G_r −−−→^br G_r−1 −−−→^br−1 . . . −−−→^b1 G₀ =P_n

for the sequence ofp-coverings of n-semi-graphs withp-rank induced by Φ. Note that for eachj = 0, . . . , r,G_j is aG/G_j-covering over P_n. By the construction of G^∗_j[p_i], we have

Φ_G∗[pi]:G^∗[p_i] =G^∗_r[p_i] −−−→^b∗r[pi] G^∗_r−1[p_i] ^b

∗r−1[pi]

−−−−→ . . . ^b

∗1[pi]

−−−→ P_n.

is a sequence of p-coverings of n-semi-graphs with p-rank. Thus, G^∗[p_i] can be regarded as a G-covering over P_n.

Note that by the construction ofG^∗[p_i], we see thatE^Φ_j^G =E^Φ_j^G∗[pi]for eachj = 1, . . . , r.

Operator II:Let us define a G-coveringG^⋆[pi] overPn. For anypi ∈v(Pn), letvi be an element of β_G⁻¹(p_i), I_vi the inertia group ofv_i. Since G is a abelian group, we may write {v_i^u}u∈G/D_vi for β_G⁻¹(p_i), and {v_i^u}u∈G/D_vi admits an natural action of G on the index set G/Dvi. We define a new semi-graph G^⋆[pi] as follows. If ♯β_G⁻¹(pi) = ♯(G/Ivi), we define G^⋆[p_i] to beG. If ♯β_G⁻¹(p_i)̸=♯(G/I_vi), we haveβ_G⁻¹(bⁱ_l) = {b^i,u,s,t_l }u∈G/D_vi,s∈I_vi^r /I_vi^l ∩I_vi^r ,t∈D^e_vi. Then β_G⁻¹(bⁱ_l) = {b^i,u,s,t_l }u∈G/D_vi,s∈I_vi^r/I_vi^l ∩I_vi^r,t∈D^e_vi admits a natural action of G as follows:

for τ ∈ G, τ(b^i,u,s,t_l ) = b^i,τ_l ^◦u,s,t if τ ̸∈ Dvi, where τ denotes the image of τ under the quotient G −→ G/D_vi, τ(b^i,u,s,t_l ) = b^i,u,τ_l ^◦s,t if τ ∈ I_v^r

i/I_v^l

i ∩ I_v^r

i, τ(b^i,u,s,t_l ) = b^i,u,s,τ_l ^◦t if τ ̸∈ I_v^l_i +I_v^r_i +I_vi, where τ denotes the image of τ under the quotient D_vi −→D^e_vi, and τ(b^i,u,s,t_l ) = b^i,u,s,t_l ifτ ∈I_vi+I_v^l

i. Similarly,β_G⁻¹(bⁱ_r) :={b^i,u,s,t_r }u∈G/D_vi,s∈I_vi^l /I_vi^l ∩I_vi^r,t∈D^e_vi also admits a natural action ofG.

Definev(G^⋆[p_i]) (resp. e(G^⋆[p_i])∪e^′(G^⋆[p_i])) to be the disjoint union (v(G)\β_G⁻¹(p_i))

⨿{v_u,t^⋆ }u∈G/D_vi,t∈D_vi^e (resp. e(G)∪e^′(G)). {v^⋆_u,t}u∈G/D_vi,t∈D_vi^e admits a natural G-action