## Local *p-Rank and Semi-Stable Reduction of Curves*

### Yu Yang

**Abstract**

In the present paper, we investigate the local*p-ranks of coverings of stable curves.*

Let*G*be a finite*p-group,f* :*Y* *−→X* a morphism of stable curves over a complete
discrete valuation ring with algebraically closed residue field of characteristic*p >*0,
*x*a singular point of the special fiber*X** _{s}* of

*X. Suppose that the generic fiberX*

*of*

_{η}*X*is smooth, and the morphism of generic fibers

*f*

*is a Galois ´etale covering with Galois group*

_{η}*G. Write*

*Y*

*for the normalization of*

^{′}*X*in the function field of

*Y*,

*ψ*:

*Y*

^{′}*−→X*for the resulting normalization morphism. Let

*y*

^{′}*∈ψ*

^{−}^{1}(x) be a point of the inverse image of

*x. Suppose that the inertia groupI*

_{y}*′*

*⊆G*of

*y*

*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*where

*I*

_{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 with*p-rank associated to vertical fibers . . . .* 27
3.3 *p-ranks of vertical fibers . . . .* 30

**4** **Appendix** **32**

**1** **Introduction and ideas**

Let*R*be a complete valuation ring with algebraically closed residue field*k*of 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 Spec*K* *−→S,*
Spec*K* *−→S, and Speck* *−→S, respectively. Let* *X* be a stable curve of genus *g** _{X}* over

*S. WriteX*

*η*,

*X*

*η*, and

*X*

*s*for the generic fiber, the geometric generic fiber, and the special fiber, respectively. Moreover, we suppose that

*X*

*is smooth over*

_{η}*η.*

Let *Y** _{η}* be a geometrically connected curve over

*η,*

*f*

*:*

_{η}*Y*

_{η}*−→X*

*a finite Galois ´etale covering over*

_{η}*η*with Galois group

*G. By replacing*

*S*by a finite extension of

*S, we may*assume that

*Y*

*admits a stable model over*

_{η}*S. Then*

*f*

*extends uniquely to a*

_{η}*G-stable*

*covering*(cf. Definition 3.3)

*f*:

*Y*

*−→X*over

*S*(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*

*of*

_{s}*Y*. If the order

*♯G*of

*G*is prime to

*p, then by the specialization theorem for log ´*etale fundamental groups,

*f*

*is an admissible covering (cf. [Y1]); thus,*

_{s}*Y*

*may be obtained by gluing together tame coverings of the irreducible components of*

_{s}*X*

*s*. On the other hand, if

*p|♯G, then*

*f*

*is not a finite morphism in general. For example, if char(K) = 0 and char(k) =*

_{s}*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 F

*p*, 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}*(x) is not finite (cf. [T], [Y2]).*

^{−1}If*f*^{−}^{1}(x) is not finite, we shall call*x*a*vertical point associated tof* and call*f*^{−}^{1}(x) the
*vertical fiber associated to* *x* (cf. Definition 3.4). In order to investigate the properties of
*Y**s*, we focus on a geometric invariant*σ(Y**s*) which is called the*p-rank ofY**s* (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 H

^{1}(Γ

_{Y}

_{s}*,*Z) (where Γ

_{Y}*denotes the dual graph of*

_{s}*Y*

*s*), the

*p-ranks of the irreducible components of*

*Y*

*s*which are finite over

*X*

*s*, 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*

*(x))*

^{−1}*is*

*always bounded by the genus of*

*Y*

*s*

*)?*

*(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.

If*x* 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 a*singular point* of *X** _{s}*. In order to explain the results obtained in the present
paper, let us introduce some notations. Write

*X*

_{1}and

*X*

_{2}for the irreducible components of

*X*

*which contain*

_{s}*x,ψ*:

*Y*

^{′}*−→X*for the normalization of

*X*in the function field of

*Y*. Let

*y*

^{′}*∈ψ*

^{−}^{1}(x) be a point in the inverse image of

*x. Write*

*I*

_{y}*′*

*⊆G*for the inertia group of

*y*

*. In order to calculate the*

^{′}*p-rank of*

*f*

^{−}^{1}(x), since

*Y /I*

_{y}*′*

*−→X*is finite ´etale over

*x,*by replacing

*X*by the stable model of the quotient

*Y /I*

_{y}*′*(note that

*Y /I*

_{y}*′*is a semi-stable curve over

*S*(cf. [R, Appendice, Corollaire])), we may assume that

*G*is equal to

*I*

_{y}*′*.

Thus, from the point of view of resolving Problem 1.1, we may assume without loss
of generality that*G*=*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} *⊂ · · · ⊂G*_{0} =*G*

be a maximal filtration of *G* (i.e., *G*_{i}*/G*_{i+1}*∼*= Z*/p*Z for *i* = 0, . . . , r*−*1). It follows from
[R, Appendice, Corollaire], that for*i*= 0, . . . , r, *Y** _{i}* :=

*Y /G*

*is a semi-stable curve over*

_{i}*S.*

Write *X*^{sst} for*Y /G* and *g* for the resulting morphism *g* :*X*^{sst}*−→X* induced by*f*. Then
we obtain a sequence of Z*/p*Z*-semi-stable coverings* (cf. Definition 3.3)

Φ* _{f}* :

*Y*=

*Y*

_{r}*−−−→*

^{d}

^{r}*Y*

_{r}

_{−}_{1}

*−−−→*

^{d}

^{r−1}*. . .*

*−−−→*

^{d}^{1}

*Y*

_{0}=

*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 each*i*= 1, . . . , r, write*ϕ** _{i}* :

*Y*

_{i}*−→Y*

_{0}for the composite morphism

*d*

_{1}

*◦· · ·◦d*

*. For simplicity, we suppose that*

_{i}*C*:=

*g*

^{−}^{1}(x)

_{red}=

*∪*

^{n}*j=1*

*P*

*, where, for each*

_{j}*j*= 1, . . . , n,

*P*

*is isomorphic to P*

_{j}^{1}and meets the other irreducible components of the special fiber

*X*

_{s}^{sst}of

*X*

^{sst}at precisely two points (i.e., a chain of P

^{1}). Thus, the

*p-rank*

*σ(f*

^{−}^{1}(x)) is equal to

*σ(ϕ*

^{−}

_{r}^{1}(C)). For each

*i*= 1, . . . , r, we define a set of subcurves of

*C*associated to Φ

*, which plays a key role in the present paper, as follows: ♦*

_{f}*E**i*^{Φ}* ^{f}* :=

*ϕ*

*(the ´etale locus of*

_{i}*d*

_{i}*|*

_{ϕ}

^{−1}*i* (C)red :*ϕ*^{−}_{i}^{1}(C)_{red} *−→ϕ*^{−}_{i}_{−}^{1}_{1}(C)_{red})*⊂C.*

We shall call*E**i*^{Φ}* ^{f}* the

*i-th ´etale-chain associated to*Φ

*and call the disjoint union*

_{f}*E*

^{Φ}

*:=⨿*

^{f}*i*

*E**i*^{Φ}^{f}

the *´etale-chain associated to* Φ* _{f}*. For each connected component

*E*of

*E*

_{i}^{Φ}

*, we use the notation*

^{f}*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* *E**i* *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*/p*Z-stable covering *f* :
*Y* *−→X*such that there exists a singular vertical point*x*such that the*p-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 for*σ(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 fiber

*Y*

*of*

_{s}*Y*such that the following conditions are satisfied: (i)

*ϕ*

*(V*

_{r}*) =*

_{i}*P*

*if*

_{i}*i*= 1, . . . , n; (ii)

*ϕ*

*(V*

_{r}_{0}) =

*X*

_{1}and

*ϕ*

*(V*

_{r}*) =*

_{n+1}*X*

_{2}; (iii) the union

*∪*

^{n+1}*i=0*

*V*

*is a connected semi-stable subcurve of the special fiber*

_{i}*Y*

*of*

_{s}*Y*. Write

*I*

_{P}

_{i}*⊆G*for the inertia subgroup of

*V*

*. Note that since*

_{i}*G*is an abelian

*p-group,I*

_{P}*does not depend on the choices of*

_{i}*V*

*.*

_{i}If *G*is 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*_{P}_{0} *is equal to* *G, then we have*
*σ(f*^{−}^{1}(x)) =*♯(G/I*_{min})*−♯(G/I*_{P}* _{n+1}*),

*where* *I*_{min} *denotes the group* *∩*^{n+1}*i=0**I*_{P}_{i}*.*

For a*G-covering of semi-graphs withp-rank, we develop a general method to compute*
the*p-rank (cf. Theorem 2.8). As an application, we generalize Sa¨ıdi’s formula to the case*
where *G*is 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*_{P}* _{i}*)

*−*

∑*n+1*

*i=1*

*♯(G/(I*_{P}_{i}_{−}_{1} +*I*_{P}* _{i}*)) + 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 where*G*is an arbitrary *p-group. Furthermore, we can*
obtain a global *p-rank formula for the special fiberY**s* (cf. [Y5]).

The present paper contains two parts. In Section 2, we develop the theory of semi-
graphs with*p-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 with*p-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 *V*G 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. Write*v(*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 by

*e. 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*=

*{b*

_{1}

*, b*

_{2}

*}*is an element of

*e*

*(G*

^{′}*) such that*

^{′}*ζ*

_{e}^{G}(b

_{1})

*∈v*(G

*) and*

^{′}*ζ*

_{e}^{G}(b

_{2})

*̸∈v(*G

*), then we have*

^{′}*ζ*

_{e}^{G}

*(b*

^{′}_{1}) =

*ζ*

_{e}^{G}(b

_{1}) and

*ζ*

_{e}^{G}

*(b*

^{′}_{2}) =

*{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 vertices*

^{′}*v(*G\G

*) is*

^{′}*v(*G)

*\v(*G

*); (ii) The set of closed edges*

^{′}*e(*G\G

*) is*

^{′}*e(*G)

*\e(*G

*); (iii) The set of open edges*

^{′}*e*

*(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*

^{′}*ζ*

*e*

^{G\G}

*(b*

^{′}*) =*

_{i}*ζ*

_{e}^{G}(b

*) (resp.*

_{i}*ζ*

*e*

^{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 P*n* a semi-graph such that
the following conditions hold: (i) *v(*P*n*) = *{p*_{1}*, . . . , p*_{n}*}*, *e(*P*n*) = *{e*_{1,2}*, . . . , e*_{n,n}_{−}_{1}*}* and
*e** ^{′}*(P

*n*) =

*{e*0,1

*, e*

*n,n+1*

*}*; (ii)

*v(e*

*i,i+1*) =

*{p*

*i*

*, p*

*i+1*

*}*; (iii)

*v(e*0,1) =

*{p*1

*}*and

*v*(e

*n,n+1*) =

*{p*

*n*

*}*. 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

*−→*P

*n*. We shall call G a

*n-semi-*

*graph withp-rank. We shall refer to*Gas the underlying semi-graph ofG,

*σ*

_{G}as the

*p-rank*map ofG,

*β*

_{G}as the base morphism ofG, respectively. We defineP

*:= (P*

_{n}*n*

*, σ*

_{P}

_{n}*, β*

_{P}

*) as follows:*

_{n}*σ*

_{P}

*(p*

_{n}*) is equal to 0 for each*

_{i}*i*= 1, . . . , n, and

*β*

_{P}

*= id*

_{n}_{P}

*is an identity morphism of semi-graph P*

_{n}*n*. We shall call P

*n*a

*n-chain.*

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

*σ(G) :=* ∑

*v**∈**v(*G)

*σ(v) +* ∑

G*i**∈**π*0(G)

rank_{Z}H^{1}(G*i**,*Z),
where *π*_{0}(*−*) 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* G^{1} :=

(G^{1}*, σ*_{G}^{1}*, β*_{G}^{1}) and G^{2} := (G^{2}*, σ*_{G}^{2}*, β*_{G}^{2}) be two *n-semi-graphs with* *p-rank. A morphism*
betweenG^{1}andG^{2} is defined by a morphism of the underlying semi-graphs*β* :G^{1} *−→*G^{2}
such that *β*_{G}^{2} *◦β* = *β*_{G}^{1}. We use the notation b : G^{1} *−→* G^{2} to denotes the morphism
of semi-graphs with *p-rank determined by* *β* : G^{1} *−→* G^{2} 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

*−→*P

*n*.

Write *b*^{i}* _{l}* (resp.

*b*

^{i}*) for*

_{r}*ζ*

_{e}

^{−}

_{i−1,i}^{1}(p

*) (resp.*

_{i}*ζ*

_{e}

^{−}^{1}

*i,i+1*(p* _{i}*)). For any element

*v*

_{i}*∈β*

_{G}

^{−}^{1}(p

*), write*

_{i}*b*

*(v*

_{l}*) (resp.*

_{i}*b*

*(v*

_{r}*)) for the set*

_{i}*{b* *∈b(v** _{i}*)

*|*

*β*

_{G}(b) =

*b*

^{i}

_{l}*}*(resp.

*{b∈b(v*

*)*

_{i}*|*

*β*

_{G}(b) =

*b*

^{i}

_{r}*}*).

**Definition 2.3.** Letb : G^{1} := (G^{1}*, σ*_{G}^{1}*, β*_{G}^{1})*−→*G^{2} := (G^{2}*, σ*_{G}^{2}*, β*_{G}^{2}) be a morphism of
*n-semi-graphs with* *p-rank,* *β* the underlying morphism of b,*e* *∈e(*G^{1})*∪e** ^{′}*(G

^{1}) an edge,

*v*

_{1}a vertex of G

^{1}contained in

*β*

_{G}

*1*

^{−}^{1}(p

*), and*

_{i}*v*

_{2}:=

*β(v*

_{1})

*∈β*

_{G}

*2*

^{−}^{1}(p

*) the image of*

_{i}*v*

_{1}.

(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 *v*_{1} *∈β*_{G}* ^{−}*1

^{1}(p

*) if one of the following*

_{i}´

etale types holds:

(Type-I) *♯β*^{−}^{1}(v2) =*p* and *σ*_{G}^{1}(v1) =*σ*_{G}^{2}(v2);

(Type-II) *♯β*^{−}^{1}(v_{2}) = 1, *♯b** _{l}*(v

_{1}) =

*p♯b*

*(v*

_{l}_{2}),

*♯b*

*(v*

_{r}_{1}) =

*p♯b*

*(v*

_{r}_{2}), and

*σ*

_{G}

^{1}(v

_{1})

*−*1 =

*p(σ*

_{G}

^{2}(v

_{2})

*−*1);

(Type-III) If *♯β*^{−}^{1}(v_{2}) = 1, *♯b** _{l}*(v

_{1}) =

*♯b*

*(v*

_{l}_{2}),

*♯b*

*(v*

_{r}_{1}) =

*p♯b*

*(v*

_{r}_{2}), and

*σ*

_{G}

^{1}(v

_{1})

*−*1 =

*p(σ*

_{G}

^{2}(v

_{2})

*−*1) + (♯b

*(v*

_{l}_{1}))(p

*−*1);

(Type-IV) *♯β*^{−}^{1}(v_{2}) = 1, *♯b** _{l}*(v

_{1}) =

*p♯b*

*(v*

_{l}_{2}),

*♯b*

*(v*

_{r}_{1}) =

*♯b*

*(v*

_{r}_{2}), and

*σ*

_{G}

^{1}(v

_{1})

*−*1 =

*p(σ*

_{G}

^{2}(v

_{2})

*−*1) + (♯b

*(v*

_{r}_{1}))(p

*−*1);

(Type-V) *♯β** ^{−1}*(v

_{2}) = 1,

*♯b*

*(v*

_{l}_{1}) =

*♯b*

*(v*

_{l}_{2}),

*♯b*

*(v*

_{r}_{1}) =

*♯b*

*(v*

_{r}_{2}), and

*σ*

_{G}

^{1}(v1)

*−*1 =

*p(σ*

_{G}

^{2}(v2)

*−*1) + (♯b

*l*(v1) +

*♯b*

*r*(v1))(p

*−*1).

(b) We shall call b*purely inseparable*at*v*_{1} *∈β*_{G}* ^{−}*1

^{1}(p

*) if*

_{i}*♯β*

^{−}^{1}(v

_{2}) = 1,

*♯b*

*(v*

_{l}_{1}) =

*♯b*

*(v*

_{l}_{2}),

*♯b** _{r}*(v

_{1}) =

*♯b*

*(v*

_{r}_{2}), and

*σ*

_{G}

^{1}(v

_{1}) =

*σ*

_{G}

^{2}(v

_{2}) hold.

(c) We shall call b a *p-covering* if the following conditions hold: (i) There exists a
Z*/p*Z-action (which may be trivial) on G^{1} (resp. a trivial Z*/p*Z-action on G^{2}), and the
underlying morphism *β* of b is compatible with the Z*/p*Z-actions. Then the natural
morphism G^{1}*/*Z*/p*Z*−→*G^{2} induced by b is an isomorphism; (ii) For any *v* *∈v(*G^{1}),b is
either*p-generically ´*etale or purely inseparable at *v; (iii) Lete∈e(*G^{1}) 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*(G

^{1}), then

*σ*

_{G}

^{1}(v) =

*σ*

_{G}

^{1}(τ(v)) holds for each

*τ*

*∈*Z

*/p*Z.

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} *⊂ · · · ⊂G*_{1} *⊂G*_{0} =*G*

an *maximal filtration* of *G* if *G*_{j}*/G*_{j+1}*∼*= Z*/p*Z for each *j* = 1, . . . , r*−*1. Suppose that
G^{1} (resp. G^{2}) admits a (resp. trivial) *G-action (which may be trivial). Then for any*
maximal filtration Φ of*G, there is a sequence of semi-graphs induced by Φ:*

G^{1} =G*r*
*β**r*

*−−−→* G*r−1* *β**r**−*1

*−−−→* *. . .* *−−−→*^{β}^{1} G0*,*

whereG*j* denotes the quotient ofG^{1} by*G** _{j}*. We shall callba

*G-covering*if for any maximal filtration Φ of

*G, there exists a set of*

*p-coverings*

*{*b

*j*: G

*j*

*−→*G

*j*

*−*1

*, j*= 1, . . . , r

*}*such that the following conditions hold: (i) the underlying morphism

*β*of bis compatible with the

*G-actions, and the natural morphism*G

^{1}

*/G−→*G

^{2}induced by

*β*is an isomorphism;

(ii) The underlying graph of G*j* is equal to G*j* for each *j* = 0, . . . , r; (iii) The underlying
morphism G*j* *−→* G*j**−*1 of b* _{j}* is equal to

*β*

*for each*

_{j}*j*= 1, . . . , r; (iv) The composite morphism b

_{1}

*◦ · · · ◦*b

*is equal tob. Then we obtain a sequence of*

_{r}*p-coverings:*

Φ_{G}^{1} :G^{1} =G_{r}*−−−→*^{b}* ^{r}* G

_{r}

_{−}_{1}

*−−−→*

^{b}

^{r−1}*. . .*

*−−−→*

^{b}

^{1}G

_{0}=G

^{2}

*.*We shall call Φ

_{G}

^{1}

*a sequence of*

*p-coverings induced by*Φ.

(f) LetGbe a*n-semi-graph withp-rank. We shall call*Ga*covering*(resp. *G-covering)*
*over* P* _{n}* if b

_{G}is a covering (resp.

*G-covering).*

(g) Let b : G^{1} *−→* G^{2} 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

*and the sequence of*

^{H}*p-coverings*

Φ^{H}_{G}1 :G^{1} =G_{r}^{b}

*H**r*

*−−−→* G_{r}_{−}_{1} ^{b}

*H**r−1*

*−−−→* *. . .* ^{b}

*H*

*−−−→*1 G_{0} =G^{2}

induced by Φ* ^{H}* such that there exists

*i*such that the underlying graph ofG

*i*is isomorphic toG

^{1}

*/H*. We writeG

^{1}

*/H*forG

*. Thus, the natural morphismb*

_{i}

^{H}_{1}

*◦· · ·◦*b

^{H}*:G*

_{i}^{1}

*/H*

*−→*G

^{2}is a covering. Then we define five subgroups of

*G*as follows:

*D** _{v}* :=

*{τ*

*∈G*

*|τ*(v) =

*v},*

*I** _{v}* := the maximal element of

*{H*

*⊆G|*G

^{1}

*−→*G

^{1}

*/H*is purely inseparable at

*v},*

*I*

_{v}*(b) :=*

^{l}*{τ*

*∈D*

_{v}*|τ*(b) =

*b*for a branch

*b∈b*

*(v)*

_{l}*}/I*

_{v}*,*

*I*_{v}* ^{r}*(b) :=

*{τ*

*∈D*

_{v}*|*

*τ(b) =b*for a branch

*b∈b*

*(v)*

_{r}*}/I*

_{v}*,*

*I*

*:=*

_{e}*{τ*

*∈G*

*|*

*τ(e) =*

*e}.*

We shall call*D** _{v}* (resp.

*I*

*,*

_{v}*I*

_{v}*(b),*

^{l}*I*

_{v}*(b),*

^{r}*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, sinceG*is an abelian

*p-group, the groupI*

_{v}*(b) (resp.*

^{l}*I*

_{v}*(b))*

^{r}does not depend on the choice of *b* *∈* *b** _{l}*(v) (resp.

*b*

*∈*

*b*

*(v)), then we denote this group briefly by*

_{r}*I*

_{v}*(resp.*

^{l}*I*

_{v}*). Define*

^{r}*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 a*G-covering over*P* _{n}*and

*v*

_{i}*∈β*

_{G}

^{−}^{1}(p

*) 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)*

_{i}*σ*_{G}(v* _{i}*)

*−*1 =

*−♯D*

_{v}

_{i}*/I*

_{v}*+*

_{i}*♯((D*

_{v}

_{i}*/I*

_{v}*)/I*

_{i}

_{v}

^{l}*)(♯I*

_{i}

_{v}

^{l}

_{i}*−*1) +

*♯((D*

_{v}

_{i}*/I*

_{v}*)/I*

_{i}

_{v}

^{r}*)(♯I*

_{i}

_{v}

^{r}

_{i}*−*1).

Let G be a *G-covering over* P*n*. 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}*−−−→*^{b}* ^{r}* G

_{r}

_{−}_{1}

*−−−→*

^{b}

^{r}

^{−}^{1}

*. . .*

*−−−→*

^{b}

^{1}G

_{0}=P

*induced by Φ. For each*

_{n}*j*= 1, . . . , r, we write

*V*

*j*

^{´}

^{et}for the set

*{v* *∈v(*G*j*)*|* b* _{j}* is ´etale at

*v},*

*E*

*j*

^{´}

^{et}for the set

*{e∈e(*G*j*)*∪e** ^{′}*(G

*j*)

*|*b

*is ´etale at*

_{j}*e}.*

Since (*V**j*^{´}^{et}*,E**j*^{´}^{et}) admits a natural structure of semi-graph induced by G*j*, we may regard
(*V*_{j}^{´}^{et}*,E*_{j}^{´}^{et}) as a sub-semi-graph of G*j*. Thus, the image *β*_{G}* _{j}*((

*V*

_{j}^{´}

^{et}

*,E*

_{j}^{´}

^{et})) can be regarded as a sub-semi-graph of P

*n*.

**Definition 2.4.** We shall call E^{Φ}*j*^{G} := *β*_{G}* _{j}*((

*V*

*j*

^{´}

^{et}

*,E*

*j*

^{´}

^{et})) (resp. the disjoint union E

^{Φ}

^{G}:=

⨿

*j*E^{Φ}*j*^{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* P*n*. We introduce two operators for G.

**Operator I:** First, let us define a *G-covering* G* ^{∗}*[p

*] over P*

_{i}*. For any*

_{n}*p*

_{i}*∈*

*v(*P

*n*), let

*v*

*be an element of*

_{i}*β*

_{G}

^{−}^{1}(p

*).*

_{i}If *♯β*_{G}^{−}^{1}(p* _{i}*) = 1 (i.e.,

*D*

_{v}*=*

_{i}*G), then we define*G

*[p*

^{∗}*] to be G; If*

_{i}*♯β*

_{G}

^{−}^{1}(p

*)*

_{i}*̸*= 1, we define a new semi-graph G

*[p*

^{∗}*] as follows.*

_{i}Define*v(*G* ^{∗}*[p

*]) (resp.*

_{i}*e(*G

*[p*

^{∗}*])*

_{i}*∪e*

*(G*

^{′}*[p*

^{∗}*])) to be the disjoint union (v(G)*

_{i}*\β*

_{G}

^{−}^{1}(p

*))⨿*

_{i}*{v*

^{∗}*}*(resp.

*e(*G)

*∪e*

*(G)).*

^{′}The collection of maps *{ζ**e*^{G}^{∗}^{[p}^{i}^{]}*}**e* is as follows: (i) For any branch *b* *̸∈ ∪*_{v}_{∈}_{β}^{−}^{1}

G (p*i*)*b(v*),
*ζ**e*^{G}^{∗}^{[p}^{i}^{]}(b) = *ζ*_{e}^{G}(b) if *b* *∈* *e* and *ζ**e*^{G}^{∗}^{[p}^{i}^{]}(b) = Ø if *b* *̸∈* *e; (ii) For any* *v* *∈* *β*_{G}^{−}^{1}(p* _{i}*) and any
branch

*b∈b(v),ζ*

*e*

^{G}

^{∗}^{[p}

^{i}^{]}(b) =

*v*

*if*

^{∗}*b*

*∈e*and

*ζ*

*e*

^{G}

^{∗}^{[p}

^{i}^{]}(b) = Ø if

*b̸∈e.*

We define a map *σ*_{G}*∗*[p*i*]:*v(*G* ^{∗}*[p

*])*

_{i}*−→*Z as follows: (i) If

*v*

^{∗}*̸*=

*v*

*∈v*(G

*[p*

^{∗}*]), then we have*

_{i}*σ*

_{G}

*∗*[p

*i*](v) :=

*σ*

_{G}(v); (ii) If

*v*=

*v*

*, then we have*

^{∗}*σ*_{G}*∗*[p*i*](v* ^{∗}*) :=

*−♯(G/I*

_{v}*) + ∑*

_{i}*v**∈**β*_{G}^{−}^{1}(p*i*)

∑

*b**∈**b** _{l}*(v)

(♯I_{v}* ^{l}*(b)

*−*1) + ∑

*v**∈**β*_{G}^{−}^{1}(p*i*)

∑

*b**∈**b**r*(v)

(♯I_{v}* ^{r}*(b)

*−*1) + 1

=*−♯(G/I**v**i*) +*♯((G/I**v**i*)/I_{v}^{l}* _{i}*)(♯I

_{v}

^{l}

_{i}*−*1) +

*♯((G/I*

*v*

*i*)/I

_{v}

^{r}*)(♯I*

_{i}

_{v}

^{r}

_{i}*−*1) + 1.

We define a morphism of semi-graphs *β*_{G}*∗*[p*i*] : G* ^{∗}*[p

*]*

_{i}*−→*P

*n*as follows: (i) For any

*v*

*∈*

*v(*G

*[p*

^{∗}*]),*

_{i}*β*

_{G}

*∗*[p

*i*](v) =

*p*

*if*

_{i}*v*=

*v*

*and*

^{∗}*β*

_{G}

*∗*[p

*i*](v) =

*β*

_{G}(v) if

*v*

*̸∈*

*β*

_{G}

^{−}^{1}(p

*); (ii) If*

_{i}*e∈e(*G

*[p*

^{∗}*])*

_{i}*∪e*

*(G*

^{′}*[p*

^{∗}*]), then we have*

_{i}*β*

_{G}

*∗*[p

*i*](e) =

*β*

_{G}(e).

Thus, the triple G* ^{∗}*[p

*] := (G*

_{i}*[p*

^{∗}*], σ*

_{i}_{G}

*∗*[p

*i*]

*, β*

_{G}

*∗*[p

*i*]) is a

*n-semi-graph with*

*p-rank.*

Moreover,G* ^{∗}*[p

*] admits a natural*

_{i}*G-action as follows: (i) the action ofG*on

*v(*G

*[p*

^{∗}*])*

_{i}*\*

*{v*

^{∗}*}*(resp.

*e(*G

*[p*

^{∗}*])*

_{i}*∪e*

*(G*

^{′}*[p*

^{∗}*])) is the action of*

_{i}*G*on

*v(*G)

*\β*

_{G}

^{−}^{1}(p

*) (resp.*

_{i}*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

*] is a*

_{i}*G-covering over*P

*. Let*

_{n}Φ :*{*1*}*=*G*_{r}*⊂G*_{r}_{−}_{1} *⊂ · · · ⊂G*_{1} *⊂G*_{0} =*G*
be an arbitrary maximal filtration of *G. Write*

Φ_{G} :G=G_{r}*−−−→*^{b}* ^{r}* G

_{r}

_{−}_{1}

*−−−→*

^{b}

^{r−1}*. . .*

*−−−→*

^{b}

^{1}G

_{0}=P

_{n}for the sequence of*p-coverings of* *n-semi-graphs withp-rank induced by Φ. Note that for*
each*j* = 0, . . . , r,G* _{j}* is a

*G/G*

*-covering over P*

_{j}*. By the construction of G*

_{n}

^{∗}*[p*

_{j}*], we have*

_{i}Φ_{G}*∗*[p*i*]:G* ^{∗}*[p

*] =G*

_{i}

^{∗}*[p*

_{r}*]*

_{i}*−−−→*

^{b}

^{∗}

^{r}^{[p}

^{i}^{]}G

^{∗}

_{r}

_{−}_{1}[p

*]*

_{i}^{b}

*∗**r**−*1[p*i*]

*−−−−→* *. . .* ^{b}

*∗*1[p*i*]

*−−−→* P_{n}*.*

is a sequence of *p-coverings of* *n-semi-graphs with* *p-rank. Thus,* G* ^{∗}*[p

*] can be regarded as a*

_{i}*G-covering over*P

*.*

_{n}Note that by the construction ofG* ^{∗}*[p

*], we see thatE*

_{i}^{Φ}

_{j}^{G}=E

^{Φ}

_{j}^{G∗[}

^{pi}^{]}for each

*j*= 1, . . . , r.

**Operator II:**Let us define a *G-covering*G* ^{⋆}*[p

*i*] overP

*n*. For any

*p*

*i*

*∈v(*P

*n*), let

*v*

*i*be an element of

*β*

_{G}

^{−}^{1}(p

*),*

_{i}*I*

_{v}*the inertia group of*

_{i}*v*

*. Since*

_{i}*G*is a abelian group, we may write

*{v*

_{i}

^{u}*}*

*u*

*∈*

*G/D*

*for*

_{vi}*β*

_{G}

^{−}^{1}(p

*), and*

_{i}*{v*

_{i}

^{u}*}*

*u*

*∈*

*G/D*

*admits an natural action of*

_{vi}*G*on the index set

*G/D*

*v*

*i*. We define a new semi-graph G

*[p*

^{⋆}*i*] as follows. If

*♯β*

_{G}

^{−}^{1}(p

*i*) =

*♯(G/I*

*v*

*i*), we define G

*[p*

^{⋆}*] to beG. If*

_{i}*♯β*

_{G}

^{−}^{1}(p

*)*

_{i}*̸*=

*♯(G/I*

_{v}*), we have*

_{i}*β*

_{G}

^{−}^{1}(b

^{i}*) =*

_{l}*{b*

^{i,u,s,t}

_{l}*}*

*u*

*∈*

*G/D*

_{vi}*,s*

*∈*

*I*

_{vi}

^{r}*/I*

_{vi}

^{l}*∩*

*I*

_{vi}

^{r}*,t*

*∈*

*D*

^{e}*. Then*

_{vi}*β*

_{G}

^{−}^{1}(b

^{i}*) =*

_{l}*{b*

^{i,u,s,t}

_{l}*}*

*u*

*∈*

*G/D*

_{vi}*,s*

*∈*

*I*

_{vi}

^{r}*/I*

_{vi}

^{l}*∩*

*I*

_{vi}

^{r}*,t*

*∈*

*D*

^{e}*admits a natural action of*

_{vi}*G*as follows:

for *τ* *∈* *G,* *τ(b*^{i,u,s,t}* _{l}* ) =

*b*

^{i,τ}

_{l}

^{◦}*if*

^{u,s,t}*τ*

*̸∈*

*D*

*v*

*i*, where

*τ*denotes the image of

*τ*under the quotient

*G*

*−→*

*G/D*

_{v}*,*

_{i}*τ*(b

^{i,u,s,t}*) =*

_{l}*b*

^{i,u,τ}

_{l}

^{◦}*if*

^{s,t}*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}

^{◦}*if*

^{t}*τ*

*̸∈*

*I*

_{v}

^{l}*+*

_{i}*I*

_{v}

^{r}*+*

_{i}*I*

_{v}*, where*

_{i}*τ*denotes the image of

*τ*under the quotient

*D*

_{v}

_{i}*−→D*

^{e}

_{v}*, and*

_{i}*τ*(b

^{i,u,s,t}*) =*

_{l}*b*

^{i,u,s,t}*if*

_{l}*τ*

*∈I*

_{v}*+*

_{i}*I*

_{v}

^{l}*i*. Similarly,*β*_{G}^{−}^{1}(b^{i}* _{r}*) :=

*{b*

^{i,u,s,t}

_{r}*}*

*u*

*∈*

*G/D*

_{vi}*,s*

*∈*

*I*

_{vi}

^{l}*/I*

_{vi}

^{l}*∩*

*I*

_{vi}

^{r}*,t*

*∈*

*D*

^{e}*also admits a natural action of*

_{vi}*G.*

Define*v*(G* ^{⋆}*[p

*]) (resp.*

_{i}*e(*G

*[p*

^{⋆}*])*

_{i}*∪e*

*(G*

^{′}*[p*

^{⋆}*])) to be the disjoint union (v(G)*

_{i}*\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}*admits a natural*

^{e}*G-action*