Raynaud-Tamagawa Theta Divisors andNew-ordinariness of Ramified Coverings of Curves

Yu Yang

Abstract

In this paper, we study coverings of curves in positive characteristic. Let X• =(X,DX) and Y • = (Y,DY ) be pointed stable curves over an algebraically closedfield k of characteristic p > 0. Suppose that X• is a generic pointed stable curve oftype (gX , nX). Let m be a natural number and Y \ DY → X \ DX a cyclic tamecovering (or equivalently, Y • → X• a cyclic admissible covering) with Galois groupZ/mZ. We give a necessary and sufficient condition for the ordinariness of Y . Thisresult generalizes a result of Nakajima concerning the ordinariness of cyclic etalecovering of generic stable curve to the case of tamely ramified coverings.

Keywords: pointed stable curve, admissible covering, generalized Hasse-Wittinvariant, new-ordinary, Raynaud-Tamagawa theta divisor, positive characteristic.

Mathematics Subject Classification: Primary 14H30; Secondary 14F35, 14G32.

Contents

1 Introduction 1

2 Preliminaries 42.1 Admissible coverings and admissible fundamental groups . . . . . . . . . . 42.2 Generalized Hasse-Witt invariants of cyclic admissible coverings . . . . . . 62.3 Raynaud-Tamagawa theta divisors . . . . . . . . . . . . . . . . . . . . . . 9

3 New-ordinariness of cyclic admissible coverings of generic curves 13

4 Applications 23

1 Introduction

In the present paper, we study coverings of curves in positive characteristic. Let

X• = (X,D)

be a smooth pointed stable curve of topological type or type, for short, (gX , nX) over analgebraically closed field k of characteristic p > 0, where X denotes the underlying curve,DX denotes the set of marked points, gX denotes the genus of X, and nX denotes the

1

cardinality #DX of DX . We put UXdef= X \DX . Moreover, by choosing a suitable base

point of UX , we have the tame fundamental group πt1(UX) of UX .

The structure of maximal prime-to-p quotient πt1(UX)

p′ of πt1(UX) is well-known, which

is isomorphic to the pro-prime-to-p completion of the following group (cf. [G])

⟨a1, . . . , agX , b1, . . . , bgX , c1, . . . , cnX|

gX∏i=1

[ai, bi]

nX∏j=1

cj = 1⟩.

On the other hand, the structure of πt1(UX) is very mysterious. In fact, some developments

of F. Pop-M. Saıdi, M. Raynaud, A. Sarashina, A. Tamagawa, and the author (cf. [PoSa],[R2], [Sa], [T1], [T2], [T3], [T4], [Y1], [Y2], [Y3], [Y5], [Y6]) showed evidence for verystrong anabelian phenomena for curves over algebraically closed fields of characteristicp > 0. In this situation, the Galois group of the base field is trivial, and the etale (ortame) fundamental group coincides with the geometric fundamental group, thus in a totalabsence of a Galois action of the base field. This kind of anabelian phenomena go beyondGrothendieck’s anabelian geometry, and shows that the tame fundamental group of asmooth pointed stable curve over an algebraically closed field must encode“moduli” ofthe curve. This is the reason that we do not have an explicit description of the tamefundamental group of any smooth pointed stable curve in positive characteristic. Notethat since all the tame coverings in positive characteristic can be lifted to characteristic0, we obtain that πt

1(UX) is topologically finitely generated. Then the isomorphism classof πt

1(UX) is determined by the set of finite quotients of πt1(UX) (cf. [FJ, Proposition

16.10.6]).Furthermore, the theory developed in [T3] and [Y2] implies that the isomorphism

class of UX as a scheme can possibly be determined by not only the isomorphism class ofπt1(UX) as a profinite group but also the isomorphism class of the maximal pro-solvable

quotient of πt1(UX). Then we may ask the following question:

Which finite solvable group can appear as a quotient of πt1(UX)?

Let N ⊆ πt1(UX) be an arbitrary open normal subgroup and X•

N = (XN , DXN) the

pointed stable curve of type (gXN, nXN

) over k corresponding to N . We have an importantinvariant σXN

associated to XN (or N) which is called p-rank (or Hasse-Witt invariant,see Definition 2.3). Roughly speaking, σXN

controls the finite quotients of πt1(UX) which

are extensions of the group πt1(UX)/N by p-groups. Since the structures of maximal

prime-to-p quotients of tame fundamental groups have been known, in order to solve thequestion mentioned above, we need compute the p-rank σXH

when πt1(UX)/N is abelian. If

πt1(UX)/N is a p-group, then σXN

can be computed by applying the Deuring-Shafarevichformula (cf. [C]). Moreover, the Deuring-Shafarevich formula implies that, to computeσXN

, we only need to assume that πt1(UX)/N is a prime-to-p group.

The situation of σXNis very complicated if πt

1(UX)/N is not a p-group. If X• isarbitrary smooth pointed stable curve over k, σXN

cannot be computed explicitly ingeneral (cf. [R1], [T3], [Y4]). Suppose that nX = 0, and that X• is a curve correspondingto a geometric generic point of the moduli space (i.e., a generic pointed stable curve).The following result proved by S. Nakajima (cf. [N]).

2

Theorem 1.1. We maintain the notation introduced above. Suppose that πt1(UX)/N is

a cyclic group, that nX = 0 (i.e., X = UX), and that X• is generic. Then we have thatσXN

= gXN(i.e., X•

N is ordinary).

Suppose that nX = 0. The computations of σXNare much more difficult than the case

where nX = 0. Note that Nakajima’s result mentioned above does not hold in general

when UXN→ UX is a tamely ramified covering, where UXN

def= XN \ DXN

. Then mainresult of the present paper is as follows, which generalizes Nakajima’s result to the caseof tamely ramified coverings, and which gives a necessary and sufficient condition for theordinariness of X•

N when πt1(UX)/N is a cyclic group (see Theorem 3.6 for precise form).

Theorem 1.2. We maintain the notation introduced above. Suppose that πt1(UX)/N ∼=

Z/prmZ is a cyclic group, and that X• is generic, where (m, p) = 1. Let t ∈ N be a natural

number such that m|(pt − 1), m′ def= (pt − 1)/m, D the ramification divisor associated to

the cyclic tame covering UXN→ UX induced by N ↪→ πt

1(UX), and D′ def= m′D. Then

we have that σXN= gXN

if and only if D′(im′), i ∈ {1, . . . ,m − 1}, is Frobenius-stable(cf. Definition 2.4 and Definition 3.4 for the definitions of D′(im′) and Frobenius-stable,respectively).

By applying Theorem 1.2, it is easy to construct prime-to-p cyclic new-ordinary (cf.Definition 2.3) tame covering of generic curves (cf. Proposition 4.1). On the other hand,we apply Theorem 1.2 to a certain inverse Galois problem for X•. Let G′ be a finite groupwhich is an extension of a group H ′ with prime-to-p order and a p-group P ′. If nX = 0,A. Pacheco and K. Stevenson gave a necessary and sufficient condition for whether or notan embedding problem (πt

1(UX) → G′, G′ ↠ H ′) has a solution (cf. [PaSt, Theorem 1.3]).Moreover, I. Bouw generalized this result to the case where nX ≥ 0 (cf. [B, Proposition 2.4and Proposition 2.5]). Roughly speaking, the criterion obtained of Pacheco-Stevenson andBouw is that the product of the multiplicity and the degree of every irreducible character ofH ′ is less than the generalized Hasse-Witt invariant associated to the irreducible characterof H ′. But, as we mentioned above, the generalized Hasse-Witt invariants cannot becomputed explicitly in general. On the other hand, by applying Nakajima (cf. Theorem1.1) and B. Zhang’s (cf. [Z]) results concerning the ordinariness of abelian etale coveringsof projective generic curves, Pacheco and Stevenson obtained a numerical criterion forwhether or not an embedding problem (πt

1(UX) → G′, G′ ↠ H ′) has a solution when H ′

is abelian (cf. [PaSt, Theorem 7.4]). When H ′ is cyclic, by applying Theorem 1.2, wegeneralize the numerical criterion of Pacheco-Stevenson to the case of ramified coverings(cf. Proposition 4.2).

The present paper is organized as follows. In Section 2, we recall some definitionsand properties of admissible coverings, generalized Hasse-Witt invariants, and Raynaud-Tamagawa theta divisors. In Section 3, we study the new-ordinariness of prime-to-p cyclicadmissible coverings of generic curves by using the theory of Raynaud-Tamagawa thetadivisors and prove our main theorem. In Section 4, we give two applications of the maintheorem.

Acknowledgements

3

This work was supported by the Research Institute for Mathematical Sciences (RIMS),an International Joint Usage/Research Center located in Kyoto University.

2 Preliminaries

2.1 Admissible coverings and admissible fundamental groups

In this subsection, we recall some definitions and results concerning admissible fundamen-tal groups which will be used in the present paper.

Definition 2.1. Let G def= (v(G), eop(G) ∪ ecl(G), {ζGe }e∈eop(G)∪ecl(G)) be a semi-graph (cf.

[Y4, Definition 2.1]). Here, v(G), eop(G), ecl(G), and {ζGe }e∈eop(G)∪ecl(G) denote the set ofvertices of G, the set of closed edges of G, the set of open edges of G, and the set of

coincidence maps of G, respectively. Note that, for each e ∈ eop(G)∪ ecl(G), edef= {b1e, b2e}

is a set of cardinality 2. Then e is a closed edge if ζGe (e) ⊆ v(G), and e is an open edge ifζGe (e) = {ζGe (e) ∩ v(G), {v(G)}}.

In the present paper, letX• = (X,DX)

be a pointed stable curve over an algebraically closed field k of characteristic p > 0, whereX denotes the underlying curve, DX denotes the set of marked points, gX denotes thegenus of X, and nX denotes the cardinality #DX of DX . We shall say that (gX , nX) is thetopological type (or type for short) of X•. Write ΓX• for the dual semi-graph of X• and

rXdef= dimQ(H

1(ΓX• ,Q)) for the Betti number of the semi-graph ΓX• . Let v ∈ v(ΓX•) ande ∈ eop(ΓX•) ∪ ecl(ΓX•). We write Xv for the irreducible component of X correspondingto v, write xe for the node of X corresponding to e if e ∈ ecl(ΓX•), and write xe for the

marked point of X corresponding to e if e ∈ eop(ΓX•). Moreover, write Xv for the smooth

compactification of UXv

def= Xv \ Xsing

v , where (−)sing denotes the singular locus of (−).We define a smooth pointed stable curve of type (gv, nv) over k to be

X•v = (Xv, DXv

def= (Xv \ UXv) ∪ (DX ∩Xv)).

LetMgX ,nX ,Z be the moduli stack of pointed stable curves of type (gX , nX) over SpecZ.We shall say that X• is generic if X• is a curve corresponding to a geometric generic pointof MgX ,nX ,Z × k. In particular, X• is smooth over k. Moreover, we shall say that X• is

a component-generic pointed stable curve over k if X•v , v ∈ v(ΓX•), is a generic pointed

stable curve of type (gv, nv) over k.

Definition 2.2. Let Y • = (Y,DY ) be a pointed stable curve over k, f • : Y • → X• amorphism of pointed stable curves over k, and f : Y → X the morphism of underlyingcurves induced by f •.

We shall say f • a Galois admissible covering over k (or Galois admissible covering forshort) if the following conditions are satisfied: (i) There exists a finite groupG ⊆ Autk(Y

•)such that Y •/G = X•, and f • is equal to the quotient morphism Y • → Y •/G. (ii) Foreach y ∈ Y sm \ DY , f is etale at y, where (−)sm denotes the smooth locus of (−). (iii)

4

For any y ∈ Y sing, the image f(y) is contained in Xsing, where (−)sing denotes the set ofsingular points of (−). (iv) For each y ∈ Y sing, we write Dy ⊆ G for the decompositiongroup of y and #Dy for the cardinality of Dy. Then we have that (#Dy, char(k)) = 1, andthat the local morphism between two nodes induced by f may be described as follows:

OX,f(y)∼= k[[u, v]]/uv → OY,y

∼= k[[s, t]]/stu 7→ s#Dy

v 7→ t#Dy .

Moreover, we have that τ(s) = ζ#Dys and τ(t) = ζ−1#Dy

t for each τ ∈ Dy, where ζ#Dy is

a primitive #Dy-th root of unit, and #(−) denotes the cardinality of (−). (v) The localmorphism between two marked points induced by f may be described as follows:

OX,f(y)∼= k[[a]] → OY,y

∼= k[[b]]a 7→ bm,

where (m, char(k)) = 1 (i.e., a tamely ramified extension).Moreover, we shall say f • an admissible covering if there exists a morphism of pointed

stable curves h• : W • → Y • over k such that the composite morphism f • ◦h• : W • → X•

is a Galois admissible covering over k. We shall say an admissible covering f • etale if fis an etale morphism.

Let Z• be a disjoint union of finitely many pointed stable curves over k. We shall saya morphism f •

Z : Z• → X• over k multi-admissible covering if the restriction of f •Z to each

connected component of Z• is admissible. For any category C , we write Ob(C ) for theclass of objects of C , and write Hom(C ) for the class of morphisms of C . We denote by

Covadm(X•)def= (Ob(Covadm(X•)),Hom(Covadm(X•)))

the category which consists of the following data: (i) Ob(Covadm(X•)) consists of anempty object and all the pairs (Z•, f •

Z : Z• → X•), where Z• is a disjoint union of finitelymany pointed stable curves over k, and f •

Z is a multi-admissible covering over k; (ii) forany (Z•, f •

Z), (Y•, f •

Y ) ∈ Ob(Covadm(X•)), we define

Hom((Z•, f •Z), (Y

•, f •Y ))

def= {g• ∈ Homk(Z

•, Y •) | f •Y ◦ g• = f •

Z},

where Homk(Z•, Y •) denotes the set of k-morphisms of pointed stable curves. It is well

known that Covadm(X•) is a Galois category. Thus, by choosing a base point x ∈ Xsm\DX ,we obtain a fundamental group πadm

1 (X•, x) which is called the admissible fundamentalgroup of X•. For simplicity of notation, we omit the base point and denote the admissiblefundamental group by

ΠX• .

Remark 2.2.1. Suppose that X• is smooth over k. By the definition of admissiblefundamental groups, the admissible fundamental group of X• is naturally isomorphic tothe tame fundamental group of X \DX .

5

Definition 2.3. Let Z• be a disjoint union of finitely many pointed stable curves over k.We define the p-rank (or Hasse-Witt invariant) of Z• to be

σZdef= dimFp(H

1et(Z,Fp)).

We shall say that Z• is ordinary ifgZ = σZ ,

where gZdef= dimk(H

1(Z,OZ)). Moreover, let Z• → X• is a multi-admissible coveringover k. We shall say that the admissible covering Z• → X• is new-ordinary if

gZ − gX = σZ − σX ,

where σX denotes the p-rank of X•. Note that if X• is ordinary, then Z• → X• isnew-ordinary if and only if Z• is ordinary.

Remark 2.3.1. Note that we have

σX =∑

v∈v(ΓX• )

σXv+ rX .

Then X• is ordinary if and only if X•v , v ∈ v(ΓX•), is ordinary. Let g• : Z• → X• is a

multi-admissible covering over k and g•v : Z•v → X•

v , v ∈ v(ΓX•), the admissible covering

over k induced by g•, where the underlying curve of Z•v is the normalization of g−1(Xv).

Then g• is new-ordinary if and only if g•v is new-ordinary for each v ∈ v(ΓX•).

2.2 Generalized Hasse-Witt invariants of cyclic admissible cov-erings

In this subsection, we recall some notation concerning generalized Hasse-Witt invariantsof cyclic admissible coverings.

We maintain the notation introduced in Section 2.1, and let X• = (X,DX) be apointed stable curve of type (gX , nX) over k, and ΠX• the admissible fundamental groupof X•. Let m be an arbitrary positive natural number prime to p and µm ⊆ k× the groupof mth roots of unity. Fix a primitive mth root ζ, we may identify µm with Z/mZ viathe map ζ i 7→ i. Let α ∈ Hom(Πab

X• ,Z/mZ). We denote by X•α = (Xα, DXα) the Galois

multi-admissible covering with Galois group Z/mZ corresponding to α. Write FXα for theabsolute Frobenius morphism on Xα. Then there exists a decomposition (cf. [Se, Section9])

H1(Xα,OXα) = H1(Xα,OXα)st ⊕H1(Xα,OXα)

ni,

where FXα is a bijection on H1(Xα,OXα)st and is nilpotent on H1(Xα,OXα)

ni. Moreover,we have

H1(Xα,OXα)st = H1(Xα,OXα)

FXα ⊗Fp k,

where (−)FXα denotes the subspace of (−) on which FXα acts trivially. Then Artin-Schreiertheory implies that we may identify

Hαdef= H1

et(Xα,Fp)⊗Fp k

6

with the largest subspace of H1(Xα,OXα) on which FXα is a bijection.The finite dimensional k-vector spaces Hα is a finitely generated k[µm]-module induced

by the natural action of µm on Xα. We have the following canonical decomposition

Hα =⊕

i∈Z/mZ

Hα,i,

where ζ ∈ µm acts on Hα,i as the ζ i-multiplication. We define

γα,idef= dimk(Hα,i), i ∈ Z/mZ.

These invariants are called generalized Hasse-Witt invariants (cf. [N]) of the cyclic multi-admissible covering X•

α → X•. Moreover, we shall say that γα,1 is the first generalizedHasse-Witt invariant of the cyclic multi-admissible covering X•

α → X•. Note that thedecomposition above implies that

dimk(Hα) =∑

i∈Z/mZ

γα,i.

In particular, if Xα is connected, then dimk(Hα) = σXα .We write Z[DX ] for the group of divisors whose supports are contained in DX . Note

that Z[DX ] is a free Z-module with basis DX . We define

c′m : Z/mZ[DX ]def= Z[DX ]⊗ Z/mZ → Z/mZ, D mod m 7→ deg(D) mod m.

Then ker(c′m) can be regarded as a subset of (Z/mZ)∼[DX ], where (Z/mZ)∼ denotesthe set {0, 1, . . . ,m − 1}, and (Z/mZ)∼[DX ] denotes the subset of Z[DX ] consisting ofthe elements whose coefficients are contained in (Z/mZ)∼. We denote by Z/mZ[DX ]

0

the kernel of c′m and by (Z/mZ)∼[DX ]0 the subset of (Z/mZ)∼[DX ] corresponding to

Z/mZ[DX ]0 under the natural bijection (Z/mZ)∼[DX ]

∼→ Z/mZ[DX ]. Note that, foreach D ∈ (Z/mZ)∼[DX ]

0, we have m|deg(D). Then

deg(D) = s(D)m

for some integer s(D) such that

0 ≤ s(D) ≤{

0, if nX ≤ 1,nX − 1, if nX ≥ 2.

Let X•,∗ = (X∗, DX∗) → X• be a universal admissible covering corresponding to ΠX• .For each e ∈ ecl(ΓX•)∪eop(ΓX•), write xe for the marked point corresponding to e, and letxe∗ be a point of the inverse image of xe in DX∗ . Write Ie∗ ⊆ ΠX• for the inertia subgroupof xe∗ . Note that Ie∗ is isomorphic to Z(1)p′ , where (−)p

′denotes the maximal prime-to-p

quotient of (−). Suppose that xe is contained in Xv. Then we have an injection

ϕe∗ : Ie∗ ↪→ ΠabX•

induced by the composition of (outer) injective homomorphisms Ie∗ ↪→ ΠX•v↪→ ΠX• ,

where ΠX•vdenotes the admissible fundamental group of X•

v . Since the image of ϕe∗

7

depends only on e, we may write Ie for the image ϕe∗(Ie∗). Moreover, the specializationtheorem of the maximal prime-to-p quotients of admissible fundamental groups of pointedstable curves (cf. [V, Theoreme 2.2 (c)]) implies that, there exists a generator [se] of Iefor each e ∈ eop(ΓX•) such that the following holds∑

e∈eop(ΓX• )

[se] = 0

in ΠabX• .

Definition 2.4. We maintain the notation introduced above.(i) We put

Dαdef=

∑e∈eop(ΓX• )

α([se])xe, α ∈ Hom(ΠabX• ,Z/mZ).

Note that we haveDα ∈ (Z/mZ)∼[DX ]0. On the other hand, for eachD ∈ (Z/mZ)∼[DX ]

0,we denote by

RevadmD (X•)def= {α ∈ Hom(Πab

X• ,Z/mZ) | Dα = D}.Moreover, we put

γ(α,D)def= γα,1.

(ii) Let Q ∈ Z[DX ] be an arbitrary effective divisor on X and m an arbitrary naturalnumber. We put

[Q/m]def=

∑x∈DX

[ordx(Q)/m]x,

which is an effective divisor on X. Here [(−)] denotes the maximum integer which is lessthan or equal to (−).

(iii) Let t ∈ N be an arbitrary positive natural number, ndef= pt − 1, and

u =t−1∑j=0

ujpj, u ∈ {0, . . . , n},

the p-adic expansion with uj ∈ {0, . . . , p − 1}. We identify {0, . . . , t − 1} with Z/tZnaturally. Then {0, . . . , t − 1} admits an additional structure induced by the naturaladditional structure of Z/tZ. We put

u(i) def=

t−1∑j=0

ui+jpj, i ∈ {0, . . . , t− 1}.

Let D ∈ (Z/nZ)∼[DX ]0. We put

D(i) def=

∑x∈DX

(ordx(D))(i)x, i ∈ {0, 1, . . . , t− 1},

which is an effective divisor on X. Moreover, for each j ∈ Z, we put

D(j)def= jD − n[jD/n].

Remark 2.4.1. We maintain the notation introduced in Definition 2.4 (iii). Then wehave

D(pt−i) = D(i), i ∈ {0, . . . , t− 1}.

8

2.3 Raynaud-Tamagawa theta divisors

In this subsection, we recall some notation and results concerning theta divisors introducedby Raynaud and Tamagawa (see also [R1] and [T3, Section 2]).

We maintain the notation introduced in Section 2.2. Moreover, in the present subsec-tion, we suppose that X• is smooth over k. The generalized Hasse-Witt invariants can bealso described in terms of line bundles and divisors. Let m ∈ N be an arbitrary naturalnumber prime to p. We denote by Pic(X) the Picard group of X. Consider the followingcomplex of abelian groups:

Z[DX ]am→ Pic(X)⊕ Z[DX ]

bm→ Pic(X),

where am(D) = ([OX(−D)],mD), bm(([L], D)) = [Lm ⊗OX(−D)]. We denote by

PX•,mdef= ker(bm)/Im(am)

the homology group of the complex. Moreover, we have the following exact sequence

0 → Pic(X)[m]a′m→ PX•,m

b′m→ Z/mZ[DX ]c′m→ Z/mZ,

where (−)[m] means the m-torsion subgroup of (−), and

a′m([L]) = ([L], 0) mod Im(am),

b′m(([L], D)) mod Im(am)) = D mod m,

c′m(D mod m) = deg(D) mod m.

We shall definePX•,m

to be the inverse image of (Z/mZ)∼[DX ]0 ⊆ (Z/mZ)∼[DX ] ⊆ Z[DX ] under the projection

ker(bm) → Z[DX ]. It is easy to see that PX•,m and PX•,m are free Z/mZ-groups withrank 2gX + nX − 1 if nX = 0 and with rank 2gX if nX = 0. Moreover, [T3, Proposition3.5] implies that

PX•,m∼= PX•,m

∼= Hom(ΠabX• ,Z/mZ).

Then every element of PX•,m induces a Galois multi-admissible covering of X• over kwith Galois group Z/mZ.

In the remainder of the present paper, we put

ndef= pt − 1,

where t ∈ N is a positive natural number. Let ([L], D) ∈ PX•,n. We put

L(j) def= L⊗j ⊗OX([jD/n])], j ∈ {1, . . . , n− 1}.

Then we have that ([L(j)], D(j)) ∈ PX•,n, and that the j-action on PX•,n is given by

([L], D) 7→ ([L(j)], D(j)).

9

Moreover, we shall denote L(j) and D(j) by L(i) and D(i), respectively, when j = pt−i,i ∈ {0, . . . , t− 1}.

On the other hand, we fix an isomorphism Ln ∼= OX(−D). Note that D is an effectivedivisor on X. We have the following composition of morphisms of line bundles

L pt→ L⊗pt = L⊗n ⊗ L ∼→ OX(−D)⊗ L ↪→ L.

The composite morphism induces a morphism ϕ([L],D) : H1(X,L) → H1(X,L). We denote

by

γ([L],D)def= dimk(

∩r≥1

Im(ϕr([L],D))).

Write αL ∈ Hom(ΠabX• ,Z/nZ) for the element corresponding to ([L], D) and FX for the

absolute Frobenius morphism on X. Then we see immediately that γαL,1 is equal to the

dimension over k of the largest subspace of H1(X,L) on which F tX

def= FX ◦ · · · ◦ FX is a

bijection. Moreover, we have

γαL,1 = dimk(H1(X,L)F t

X ⊗Fp k),

where (−)FtX denotes the subspace of (−) on which F t

X acts trivially. It is easy to checkthat

H1(X,L)F tX ⊗Fp k =

∩r≥1

Im(ϕr([L],D)).

Then we obtain that γ([L],D) = γαL,1. Moreover, we observer that DαL = D. Then weobtain that

γ([L],D) = γ(αL,D)(def= γαL,1).

Lemma 2.5. We maintain the notation introduced above. Suppose that X• is smoothover k. Then we have

γ(αL,D) ≤ dimk(H1(X,L)) =

gX , if ([L], D) = ([OX ], 0),gX − 1, if s(D) = 0,gX + s(D)− 1, if s(D) ≥ 1.

Proof. The first inequality follows from the definition of generalized Hasse-Witt invariants.The Riemann-Roch theorem implies that

dimk(H1(X,L)) = gX − 1− deg(L) + dimk(H

0(X,L))

= gX − 1 +1

ndeg(D) + dimk(H

0(X,L)) = gX − 1 + s(D) + dimk(H0(X,L)).

This completes the proof of the lemma.

Next, let us explain the Raynaud-Tamagawa theta divisors. Let Fk be the absolute

Frobenius morphism on Spec k and FX/k the relative Frobenius morphism X → X1def=

X ×k,Fkk over k and F t

kdef= Fk ◦ · · · ◦ Fk. We define

Xtdef= X ×k,F t

kk,

10

and define a morphismF tX/k : X → Xt

over k to be F tX/k

def= FXt−1/k ◦ · · · ◦ FX1/k ◦ FX/k.

Let L be a line bundle on X of degree −s(D) and Lt the pulling-back of L by thenatural morphism Xt → X. We put

BtD

def= (F t

X/k)∗(OX(D))/OXt , EDdef= Bt

D ⊗ Lt.

Write rk(ED) for the rank of ED. Then we have

χ(ED) = deg(det(ED))− (gX − 1)rk(ED).

Moreover, χ(ED) = 0 (cf. [T3, Lemma 2.3 (ii)]). In [R1], Raynaud investigated thefollowing property of the vector bundle ED on X.

Condition 2.6. We shall say that ED satisfies (⋆) if there exists a line bundle L′t of degree

0 on Xt such that

0 = min{dimk(H0(Xt, ED ⊗ L′

t)), dimk(H1(Xt, ED ⊗ L′

t))}.

Let JXt be the Jacobian variety of Xt, and LXt a universal line bundle on Xt × JXt .Let prXt

: Xt × JXt → Xt and prJXt: Xt × JXt → JXt be the natural projections. We

denote by F the coherent OXt-module pr∗Xt(ED)⊗ LXt , and by

χFdef= dimk(H

0(Xt ×k k(y),F ⊗ k(y)))− dimk(H1(Xt ×k k(y),F ⊗ k(y)))

for each y ∈ JXt , where k(y) denotes the residue field of y. Note that since prJXtis flat,

χF is independent of y ∈ JXt . Write (−χF)+ for max{0,−χF}. We denote by

ΘED ⊆ JXt

the closed subscheme of JXt defined by the (−χF)+-th Fitting ideal

Fitt(−χF )+(R1(prJXt

)∗(F)).

The definition of ΘED is independent of the choice of Lt. Moreover, for each line bundleL′′ of degree 0 on Xt, we have that [L′′] ∈ ΘED if and only if

0 = min{dimk(H0(Xt, ED ⊗ L′′)), dimk(H

1(Xt, ED ⊗ L′′))},

where [L′′] denotes the point of JXt corresponding to L′′ (cf. [T3, Proposition 2.2 (i) (ii)]).Suppose that ED satisfies (⋆). [R1, Proposition 1.8.1] implies that ΘED is algebraically

equivalent to rk(ED)Θ, where Θ is the classical theta divisor (i.e., the image of XgX−1t in

JXt). Then we have the following definition.

Definition 2.7. We shall say ΘED ⊆ JXt the Raynaud-Tamagawa theta divisor associatedto ED if ED satisfies (⋆).

11

Remark 2.7.1. We maintain the notation introduced above. Moreover, we suppose that

([L], D) ∈ PX•,n. The definition of ED implies that the following natural exact sequenceholds

0 → Lt → (F tX/k)∗(OX(D))⊗ Lt → ED → 0.

Let [I] ∈ Pic(X)[n]. Write It for the pulling-back of I by the natural morphism Xt → X.we obtain the following exact sequence

. . . → H0(Xt, ED ⊗ It) → H1(Xt,Lt ⊗ It)ϕLt⊗It→ H1(Xt, (F

tX/k)∗(OX(D))⊗ Lt ⊗ It)

→ H1(Xt, ED ⊗ It) → . . . .

Note that we have that

H1(Xt,Lt ⊗ It) ∼= H1(X,L ⊗ I),

and that

H1(Xt, (FtX/k)∗(OX(D))⊗ Lt ⊗ It) ∼= H1(X,OX(D)⊗ (F t

X/k)∗(Lt ⊗ It))

∼= H1(X,OX(D)⊗ (L ⊗ I)⊗pt) ∼= H1(X,L ⊗ I).

Moreover, it is easy to see that the homomorphism

H1(X,L ⊗ I) → H1(X,L ⊗ I)

induced by ϕLt⊗It coincides with ϕ([L⊗I],D). Thus, we obtain that if

γ(([L⊗I],D)) = dimk(H1(X,L ⊗ I))

for some line bundle [I] ∈ Pic(X)[n], then the Raynaud-Tamagawa theta divisor ΘEDassociated to ED exists (i.e., [It] ∈ ΘED).

Remark 2.7.2. Suppose that the Raynaud-Tamagawa theta divisor associated to EDexists. We see immediately that the Raynaud-Tamagawa theta divisor associated toED(i) , i ∈ {0. . . . , t− 1}, exists.

The following fundamental theorem of theta divisors was proved by Raynaud andTamagawa.

Theorem 2.8. Suppose that s(D) ∈ {0, 1}. Then the Raynaud-Tamagawa theta divisorassociated to ED exists (i.e., ED satisfies (⋆)).

Remark 2.8.1. Theorem 2.8 was proved by Raynaud if s(D) = 0 (cf. [R1, Theoreme4.1.1]), and by Tamagawa if s(D) ≤ 1 (cf. [T3, Theorem 2.5]). On the other hand, theRaynaud-Tamagawa theta divisor ΘED associated to ED does not exist in general (cf. [T3,Example 2.19]).

By applying Theorem 2.8, the following lemma was proved by Tamagawa (cf. [T3,Corollary 2.6, Lemma 2.12 (ii)]).

12

Lemma 2.9. Let X• be a smooth pointed stable curve over k.(i) Let Q ∈ Z[DX ] be an effective divisor on X of degree s(Q)n such that ordx(Q) ≤ n

for each x ∈ Supp(Q), LQ a line bundle on X of degree −s(Q), and LQ,t the pulling-back

of LQ by the natural morphism Xt → X. Let SQdef= {x ∈ X | ordx(Q) = n},

Q′ def= Q− (

∑x∈SQ

nx).

an effective divisor on X of degree s(Q′)n, LQ′ a line bundle on X of degree −s(Q′), andLQ′,t the pulling-back of LQ′ by the natural morphism Xt → X. Suppose that the Raynaud-Tamagawa theta divisor associated to Bt

Q⊗LQ,t exists. Then the Raynaud-Tamagawa thetadivisor associated to Bt

Q′ ⊗ LQ′,t exists.

(ii) Let ti, i ∈ {1, 2}, be an arbitrary positive natural number and nidef= pti − 1. Let

Qi ∈ Z[DX ] be an effective divisor on X of degree deg(Qi) = s(Qi)ni, LQia line bundle on

X of degree −s(Qi), and LQi,ti the pulling-back of LQiby the natural morphism Xti → X.

Suppose that sdef= s(Q1) = s(Q2). Let t

def= t1 + t2, n

def= n1 + pt1n2,

Qdef= Q1 + pt1Q2 ∈ Z[DX ]

an effective divisor on X of degree deg(Q) = sn, LQ a line bundle on X of degree−s, and LQ,t the pulling-back of LQ by the natural morphism Xt → X. Then theRaynaud-Tamagawa theta divisor associated to Bt

Q⊗LQ,t exists if and only if the Raynaud-

Tamagawa theta divisor associated to BtiQi

⊗ LQi,ti exists for each i ∈ {1, 2}.

3 New-ordinariness of cyclic admissible coverings of

generic curves

In this section, we discuss new-ordinariness of prime-to-p cyclic admissible coverings ofgeneric curves. We maintain the notation introduced in Section 2. Let X• be a pointedstable curve of type (gX , nX) over an algebraically closed field k of characteristic p > 0.First, we have the following result.

Lemma 3.1. Let D ∈ (Z/nZ)∼[DX ]0 and α ∈ RevadmD (X•) such that α = 0. Suppose

that n = p − 1, and that X• is a generic pointed stable curve of type (0, 3). Then theRaynaud-Tamagawa theta divisor ΘED associated to ED exists. Moreover, we have

γ([L],D) = dimk(H1(X,L)), ([L], D) ∈ PX•,n.

Proof. This follows immediately from [B, Corollary 6.8].

Let R be a discrete valuation ring with algebraically closed residue field kR, KR thequotient field of R, and KR an algebraic closure of KR. Suppose that k ⊆ KR. Let

X • = (X , DXdef= {e1, . . . , enX

})

13

be a pointed stable curve of type (gX , nX) over R. We shall write X •η = (Xη, DXη

def=

{eη,1, . . . , eη,nX}), X •

η = (Xη, DXη

def= {eη,1, . . . , eη,nX

}), X •s = (Xs, DXs

def= {es,1, . . . , es,nX

})for the generic fiber X • ×R KR of X •, the geometric generic fiber X • ×R KR of X •, andthe special fiber X • ×R kR of X •, respectively. Write ΠX •

ηand ΠX •

sfor the admissible

fundamental groups of X •η and X •

s , respectively. Then we obtain a specialization map

spR : ΠX •η↠ ΠX •

s

which is surjective.We shall say that X• admits a (DEG-A) if the following conditions hold, where

“(DEG)” means “degeneration”:

(i) The geometric generic fiber X •η of X • is KR-isomorphic to X•×kKR. Then

without loss of generality, we may identify eη,r, r ∈ {1, . . . , nX}, with xr×kKR

via this isomorphism. Note that since the admissible fundamental groups donot depend on the base fields, ΠX •

ηis naturally isomorphic to ΠX• .

(ii) X •s is a component-generic pointed stable curve over kR.

(iii) If nX ≤ 1, we have X • → SpecR is isotrivial (i.e., the image of the naturalmorphism SpecR → MgX ,nX ,kR → M gX ,nX ,kR determined by X • → SpecR is

a point, where MgX ,nX ,kRdef= MgX ,nX ,Z ×Fp kR and M gX ,nX ,kR denotes the

coarse moduli space of MgX ,nX ,kR).

(iv) If nX = 2, the underlying curve of X •s is

Xs = C ∪ P

such that the following conditions hold: (a) C is a projective curve of genusgX over kR. (b) P is kR-isomorphic to P1

kR. (c) #(C ∩ P ) = 1. (d) DXs ⊆ P .

(v) If nX ≥ 3, the underlying curve of X •s is

Xs = C ∪ (

nX−2∪v=1

Pv)

such that the following conditions hold: (a) C is an empty set when gX = 0;otherwise, C is a projective curve of genus gX over kR. (b) Pv, v ∈ {1, . . . , nX−2}, is kR-isomorphic to P1

kR. (c) If C is not empty, we have #(C ∩P1) = 1 and

C ∩ Pv = ∅ for each v ∈ {2, . . . , nX − 2}. (d) For each v ∈ {1, . . . , nX − 2},#(Pv∩Pv+1) = 1 and Pv∩Pv′ = ∅ when v′ ∈ {v−1, v, v+1}. (e) If nX = 3, wehave DXs ∩ P1 = {es,1, es,2, es,3}. (f) If nX = 4, we have DXs ∩ P1 = {es,1, es,2}and DXs ∩ P2 = {es,3, es,4}. (g) If nX ≥ 5, we have DXs ∩ P1 = {es,1, es,2},DXs ∩PnX−2 = {es,nX−1, es,nX

}, and DXs ∩Pv = {es,v+1}, v ∈ {2, . . . , nX − 3}.

Moreover, we shall say that X• admits a (DEG-B) if the following conditions hold:

14

(i) The geometric generic fiber X •η of X • is KR-isomorphic to X•×kKR. Then

without loss of generality, we may identify eη,r, r ∈ {1, . . . , nX}, with xr×kKR

via this isomorphism.

(ii) X •s is a component-generic pointed stable curve over kR.

(iii) gX ≥ 1 and nX ≥ 2.

(iv) The underlying curve of X •s is

Xs = C ∪ P

such that the following conditions hold: (a) C is a projective curve of genusgX over kR. (b) P is kR-isomorphic to P1

kR. (c) #(C ∩ P ) = 1. (d) DXs ⊆ P .

Lemma 3.2. Let D ∈ (Z/nZ)∼[DX ]0 and α ∈ RevadmD (X•) such that α = 0. Suppose

that X• is generic, and that n = p− 1. Then the Raynaud-Tamagawa theta divisor ΘEDassociated to ED exists. Moreover, we have

γ([L],D) = dimk(H1(X,L)), ([L], D) ∈ PX•,n.

Proof. Let f • : Y • = (Y,DY ) → X• be the Galois multi-admissible covering over k withGalois group Z/nZ induced by α. We note that, to verify the lemma, we only need toprove the lemma in the case where Y • is connected. Then we may assume that Y • isconnected.

Since X• is a generic curve, X• admits a (DEG-A). Furthermore, we write Qη (resp.Qs) for the effective divisor on Xη (resp. Xs) induced by D and αη ∈ RevadmQη

(X •η ) for the

element induced by α. Then we have

γ(α,D) = γ(αη ,Qη).

Suppose that X• satisfies (DEG-A)-(iii). If nX = 1 and gX = 1, then the lemma istrivial. If nX ≤ 1 and gX ≥ 2, then the lemma follows immediately from [N, Proposition4].

Suppose that X• satisfies (DEG-A)-(v). Moreover, we suppose that C = ∅, and thatnX ≥ 5. For each v ∈ {1, . . . , nX − 3}, we write

yv and zv+1

for the inverse image of Pv ∩ Pv+1 of the natural closed immersion Pv ↪→ Xs and theinverse image of Pv ∩ Pv+1 of the natural closed immersion Pv+1 ↪→ Xs, respectively. Wedefine

P •1 = (P1, DP1

def= {es,1, es,2, y1} ∪ (C ∩ P1)),

P •nX−2 = (PnX−2, DPnX−2

def= {znX−2, es,nX−1, es,nX

}),

andP •v = (Pv, DPv

def= {zv, es,v+1, yv}), v ∈ {2, . . . , nX − 3},

15

to be smooth pointed stable curves of types (0, 4), (0, 3), and (0, 3) over kR, respectively.Moreover, we define

C• = (C,DCdef= C ∩ P1)

to be a smooth pointed stable curve of type (gX , 1) over kR. Note that since C• is generic,

we have σC = gX . Let

f •η

def= f • ×k KR : Y•

η = (Yη, DYη)def= Y • ×k KR → X •

η

be the Galois admissible covering over KR with Galois group Z/nZ induced by f •, andΠY•

η⊆ ΠX •

ηthe admissible fundamental group of Y•

η . By the specialization theorem of

maximal prime-to-p quotients of admissible fundamental groups (cf. [V, Theoreme 2.2(c)]), we have

spp′

R : Πp′

X •η

∼→ Πp′

X •s,

where (−)p′denotes the maximal prime-to-p quotient of (−). Then we obtain a normal

open subgroup Πp′

Y•s

def= spp

′

R(Πp′

Y•η) ⊆ Πp′

X •s. Write ΠY•

s⊆ ΠX •

sfor the inverse image of Πp′

Y•s

of the natural surjection ΠX •s↠ Πp′

X •s. Then ΠY•

sdetermines a Galois admissible covering

f •s : Y•

s = (Ys, DYs) → X •s

over kR with Galois group Z/nZ. Write αs ∈ RevadmQs(X •

s ) for an element induced by f •s .

The structure of the maximal prime-to-p quotients of admissible fundamental groupsimplies that fs is etale over C. Then we obtain that fs is etale over C ∩ P1. Thus, fs is

etale over DC . Let Yvdef= f−1

s (Pv), v ∈ {1, . . . , nX − 2}. We put

Y •v

def= (Yv, DYv

def= f−1

s (DPv)), v ∈ {1, . . . , nX − 2}.

Then f •s induces a Galois multi-admissible covering

f •v : Y •

v → P •v , v ∈ {1, . . . , nX − 2},

over kR with Galois group Z/nZ. Write ([Lv], Qv) ∈ PP •v ,n for the element induced by

f •v for each v ∈ {1, . . . , nX − 2}. Note that since fs is etale over C ∩ P1, we have thatSupp(Q1) ⊆ {es,1, es,2, y1}. Moreover, the kR[µn]-module H1

et(Yv,Fp) ⊗ kR admits thefollowing canonical decomposition

H1et(Yv,Fp)⊗ kR =

⊕j∈Z/nZ

MYv(j),

where ζ ∈ µn acts on MYv(j) as the ζj-multiplication. Then Lemma 3.1 implies that

γ([Lv ],Qv) = dimkR(MYv(1)) = dimkR(H1(Pv,Lv)).

Let Zdef= f−1

s (C) and π0(Z) the set of connected components of Z. Then f •s induces

a Galois etale covering (not necessarily connected)

f •C : Z• = (Z,DZ

def= f−1

s (DC)) → C•

16

over kR with Galois group Z/nZ. Moreover, f •C induces an element αC ∈ Revadm0 (C•).

Suppose that #π0(Z) = n. Then we have αC = 0. The kR[µn]-module H1et(Z,Fp) ⊗ kR

admits the following canonical decomposition

H1et(Z,Fp)⊗ kR =

⊕j∈Z/nZ

MZ(j),

where ζ ∈ µn acts on MZ(j) as the ζj-multiplication. [N, Proposition 4] implies that

γ(αC ,0) = dimkR(MZ(1)) = gX − 1.

Suppose that #π0(Z) = n. Then we have αC = 0. Since C is ordinary, we obtainimmediately that

γ(αC ,0) = gX .

Write ΓY•sfor the dual semi-graph of Y•

s . The natural k[µn]-submodule

H1(ΓY•s,Fp)⊗ k ⊆ H1

et(Ys,Fp)⊗ k

admits the following canonical decomposition

H1(ΓY•s,Fp)⊗ k =

⊕j∈Z/nZ

MΓY•s(j),

where ζ ∈ µn acts on MΓY•s(j) as the ζj-multiplication. Then we see immediately that

γ(αs,Qs) = γ(αC ,0) +

nX−2∑v=1

γ([Lv ],Qv) + dimk(MΓY•s(1)) = gX + s(Dαs)− 1.

On the other hand, the kR[µn]-modules H1et(Yη,Fp) ⊗ kR and H1

et(Ys,Fp) ⊗ kR admitthe following canonical decompositions

H1et(Yη,Fp)⊗ kR =

⊕j∈Z/nZ

MYη(j)

andH1

et(Ys,Fp)⊗ kR =⊕

j∈Z/nZ

MYs(j),

respectively, where ζ ∈ µn acts on MYη(j) and MYs(j) as the ζ

j-multiplication. Moreover,we have an injection as kR[µn]-modules

H1et(Ys,Fp)⊗ kR ↪→ H1

et(Yη,Fp)⊗ kR

induced by the specialization map ΠY•η↠ ΠY•

s. Thus, we have

gX + s(Dαs)− 1 = γ(αs,Qs) = dimkR(MYs(1))

≤ γ(αη ,Qη) = dimkR(MYη(1)) ≤ gX + s(Dαη

)− 1.

17

Since s(D) = s(Dαη) = s(Dαs), we obtain that

γ([L],D) = gX + s(D)− 1 = dimk(H1(X,L)).

This completes the proof of the lemma if X• satisfies (DEG-A)-(v), C = ∅, and nX ≥ 5.By applying similar arguments to the arguments given in the proof above, one can provethe lemma immediately when X• satisfies (DEG-A)-(v) and either C = ∅ or nX ≤ 4holds.

Moreover, similar arguments to the arguments given in the proof above imply thelemma holds when X• satisfies (DEG-A)-(iv). We complete the proof of the lemma.

Lemma 3.3. Let D ∈ (Z/nZ)∼[DX ]0 be an effective divisor on X of degree s(D)n and

x ∈ DX . For each i ∈ {0, . . . , t− 1}, we put

d(i)xdef= ordx(D

(i)),

and write

d(i)x =t−1∑r=0

d(i)x,rpr

for the p-adic expansion. Then the following statements are equivalent:(i)

s(D)n = deg(D) = deg(D(i))

holds for each i ∈ {0, 1, . . . , t− 1}.(ii) ∑

x∈DX

dx,r = s(D)(p− 1)

holds for each r ∈ {0, . . . , t− 1}.(iii) ∑

x∈DX

d(i)x,r = s(D)(p− 1)

holds for each i ∈ {0, . . . , t− 1} and each r ∈ {0, . . . , t− 1}.

Proof. We see that (ii) ⇒ (iii) and (iii) ⇒ (i) follows immediately from the definition ofD(i). Let us prove (i) ⇒ (ii).

Let r ∈ {0, . . . , t− 1}. We have

d(r+1)x = dx,rp

t−1 +d(r)x − dx,r

p=

1

pd(r)x +

pt − 1

pdx,r =

1

pd(r)x +

n

pdx,r.

Note that (i) implies that

s(D)n =∑x∈DX

d(r+1)x =

∑x∈DX

d(r)x .

Then we have

s(D)n =∑x∈DX

d(r+1)x =

1

p

∑x∈DX

d(r)x +n

p

∑x∈DX

dx,r

18

=1

ps(D)n+

n

p

∑x∈DX

dx,r.

This means that ∑x∈DX

dx,r = s(D)(p− 1).

We complete the proof of the lemma.

We introduce the following condition concerning effective divisors on X.

Definition 3.4. Let t ∈ N be an arbitrary positive natural number and n = pt − 1.Let D ∈ (Z/nZ)∼[DX ]

0. We shall say that D is Frobenius-stable if one of statementsmentioned in Lemma 3.3 holds.

Proposition 3.5. Let D ∈ (Z/nZ)∼[DX ]0 and α ∈ RevadmD (X•) such that α = 0. Suppose

that X• is generic, and that D is Frobenius stable. Then the Raynaud-Tamagawa theta

divisor ΘED associated to ED exists. Moreover, for each ([L], D) ∈ PX•,n, we have

γ([L],D) = dimk(H1(X,L)) = dimk(H

1(X,L(i))) = γ([L(i)],D(i)), i ∈ {0, . . . , t− 1}.

Proof. Since D is Frobenius-stable, we have that dimk(H1(X,L)) = dimk(H

1(X,L(i)))for each i ∈ {0, . . . , t − 1}. Then to verify the proposition, it is sufficient to prove thatγ([L(i)],D(i)) = dimk(H

1(X,L(i))) holds for each i ∈ {0, . . . , t − 1}. Furthermore, it is easyto see that it is sufficient to prove that

γ([L],D) = dimk(H1(X,L)).

Suppose that gX = 0. We maintain the notation introduced in Lemma 3.3. We define

Drdef=

∑x∈DX

dx,rx, r ∈ {0, . . . , t− 1},

to be an effective divisor onX. Note that deg(Dr) = s(D)(p−1) for each r ∈ {0, . . . , t−1},and that

D =t−1∑r=0

Drpr.

Then the proposition follows immediately from Lemma 2.9 and Lemma 3.2.Suppose that gX ≥ 1 and nX ≤ 1. Then the proposition follows immediately from [N,

Proposition 4].Suppose that gX ≥ 1 and nX ≥ 2. Since X• is generic, we see that X• admits a

(DEG-B). Let f • : Y • = (Y,DY ) → X• be the Galois multi-admissible covering over kwith Galois group Z/nZ induced by α. We note that, to verify the proposition, we onlyneed to prove the proposition in the case where Y • is connected. Then we may assumethat Y • is connected. Furthermore, we write Qη (resp. Qs) for the effective divisor on Xη

(resp. Xs) induced by D and αη ∈ RevadmQη(X •

η ) for the element induced by α. Then wehave

γ(α,D) = γ(αη ,Qη).

19

We defineP • def

= (P,DPdef= DXs)

andC• = (C,DC

def= C ∩ P )

to be smooth pointed stable curves of types (0, nX + 1) and (gX , 1) over k, respectively.Let

f •η

def= f • ×k KR : Y•

η = (Yη, DYη)def= Y • ×k KR → X •

η

be the Galois admissible covering over KR with Galois group Z/nZ induced by f •, andΠY•

η⊆ ΠX •

ηthe admissible fundamental group of Y•

η . By the specialization theorem of

maximal prime-to-p quotients of admissible fundamental groups (cf. [V, Theoreme 2.2(c)]), we have

spp′

R : Πp′

X •η

∼→ Πp′

X •s,

where (−)p′denotes the maximal prime-to-p quotient of (−). Then we obtain a normal

open subgroup Πp′

Y•s

def= spp

′

R(Πp′

Y•η) ⊆ Πp′

X •s. Write ΠY•

s⊆ ΠX •

sfor the inverse image of Πp′

Y•s

of the natural surjection ΠX •s↠ Πp′

X •s. Then ΠY•

sdetermines a Galois admissible covering

f •s : Y•

s = (Ys, DYs) → X •s

over kR with Galois group Z/nZ. Write αs ∈ RevadmQs(X •

s ) for an element induced by f •s .

The structure of the maximal prime-to-p quotients of admissible fundamental groupsimplies that fs is etale over C. Then we obtain that fs is etale over C ∩ P . Thus, fs is

etale over DC . Let Yvdef= f−1

s (P ). We put

Y • def= (Y,DY

def= f−1

s (DP )).

Then f •s induces a Galois multi-admissible covering

f •P : Y • → P •,

over kR with Galois group Z/nZ. Write ([LP ], QP ) ∈ PP •,n for the element induced byf •P . Note that since fs is etale over C ∩ P , we have that Supp(QP ) ⊆ DP . Moreover, thekR[µn]-module H1

et(Y,Fp)⊗ kR admits the following canonical decomposition

H1et(Y,Fp)⊗ kR =

⊕j∈Z/nZ

MY (j),

where ζ ∈ µn acts on MY (j) as the ζj-multiplication. Then the case of g = 0 implies that

γ([LP ],QP ) = dimkR(MY (1)) = dimkR(H1(P,LP )) = s(Dαs)− 1.

Let Zdef= f−1

s (C) and π0(Z) the set of connected components of Z. Then f •s induces

a Galois etale covering (not necessarily connected)

f •C : Z• = (Z,DZ

def= f−1

s (DC)) → C•

20

over kR with Galois group Z/nZ. Moreover, f •C induces an element αC ∈ Revadm0 (C•).

Suppose that #π0(Z) = n. Then we have αC = 0. The kR[µn]-module H1et(Z,Fp) ⊗ kR

admits the following canonical decomposition

H1et(Z,Fp)⊗ kR =

⊕j∈Z/nZ

MZ(j),

where ζ ∈ µn acts on MZ(j) as the ζj-multiplication. [N, Proposition 4] implies that

γ(αC ,0) = dimkR(MZ(1)) = gX − 1.

Suppose that #π0(Z) = n. Then we have αC = 0. Since C is ordinary, we obtainimmediately that

γ(αC ,0) = gX .

Write ΓY•sfor the dual semi-graph of Y•

s . The natural k[µn]-submodule

H1(ΓY•s,Fp)⊗ k ⊆ H1

et(Ys,Fp)⊗ k

admits the following canonical decomposition

H1(ΓY•s,Fp)⊗ k =

⊕j∈Z/nZ

MΓY•s(j),

where ζ ∈ µn acts on MΓY•s(j) as the ζj-multiplication. Then we see immediately that

dimk(MΓY•s(1)) =

{0, if #π0(Z) = n,1, if #π0(Z) = 1.

Then we have

γ(αs,Qs) = γ(αC ,0) + γ([LP ],QP ) + dimk(MΓY•s(1)) = gX + s(Dαs)− 1.

On the other hand, the kR[µn]-modules H1et(Yη,Fp) ⊗ kR and H1

et(Ys,Fp) ⊗ kR admitthe following canonical decompositions

H1et(Yη,Fp)⊗ kR =

⊕j∈Z/nZ

MYη(j)

andH1

et(Ys,Fp)⊗ kR =⊕

j∈Z/nZ

MYs(j),

respectively, where ζ ∈ µn acts on MYη(j) and MYs(j) as the ζ

j-multiplication. Moreover,we have an injection as kR[µn]-modules

H1et(Ys,Fp)⊗ kR ↪→ H1

et(Yη,Fp)⊗ kR

induced by the specialization map ΠY•η↠ ΠY•

s. Thus, we have

gX + s(Dαs)− 1 = γ(αs,Qs) = dimkR(MYs(1))

21

≤ γ(αη ,Qη) = dimkR(MYη(1)) ≤ gX + s(Dαη

)− 1.

Since s(D) = s(Dαη) = s(Dαs), we obtain that

γ([L],D) = gX + s(D)− 1 = dimk(H1(X,L)).

This completes the proof of the proposition.

The main result of the present paper is as follows.

Theorem 3.6. Let m ∈ N be an arbitrary positive natural number prime to p and D ∈(Z/mZ)∼[DX ]

0. Let t ∈ N be a positive natural number such that pt = 1 in (Z/mZ)×,n

def= pt − 1, and m′ def

= n/m. Write D′ for the divisor m′D ∈ (Z/nZ)∼[DX ]0 when we

identify Z/mZ with the unique subgroup of Z/nZ of order m. Let ([L], D) ∈ PX•,m and

f • : Y • → X•

the Galois multi-admissible covering with Galois group Z/mZ induced by ([L], D). Supposethat X• is generic. Then f • is new-ordinary if and only if D′(jm′), j ∈ {1, . . . ,m − 1},is Frobenius-stable.

Proof. We see immediately that, to verify the proposition, it is sufficient to prove thetheorem when D = D′.

Since we have the following canonical isomorphism

f∗OY∼= OX ⊕

n⊕j=1

L(j),

we obtain that

gY = gX +n−1∑j=1

dimk(H1(X,L(j))).

On the other hand, we have

σY = σX +n−1∑j=1

γ([L(j)],D(j))

and γ([L(j)],D(j)) ≤ dimk(H1(X,L(j))) for each j ∈ {1, . . . , n}.

Suppose that D(j), j ∈ {1, . . . , n}, is Frobenius-stable. Then Proposition 3.5 impliesthat Y • is ordinary. Conversely, suppose that Y • is ordinary. Let j ∈ {1, . . . , n} andi ∈ {0, . . . , t − 1}. Then we have γ([L(i)(j)],D(i)(j)) = dimk(H

1(X,L(i)(j))). Moreover, wehave

γ([L(i)(j)],D(i)(j)) ≤ dimk(H1(X,L(i′)(j)))

holds for each i′ ∈ {0, . . . , t− 1}. This implies that

gX + s(D(i)(j))−1 = dimk(H1(X,L(i)(j))) = dimk(H

1(X,L(i′)(j))) = gX + s(D(i′)(j))−1

holds for each i′ ∈ {0, . . . , t− 1}. This means that the statement of Lemma 3.3 (i) holds.Then D(j), j ∈ {1, . . . , n}, is Frobenius-stable.

22

Remark 3.6.1. Let 0 ≤ σ ≤ gX be an integer, MgX ,nX ,Z the moduli stack of pointed

stable curves of type (gX , nX), MgX ,nX

def= MgX ,nX ,Z ×Z k, and Mσ

gX ,nXthe locally closed

reduced substack of MgX ,nXwhose points represent pointed stable curves with p-rank σ.

Suppose that X• be a pointed stable curve corresponding to a geometric point over ageneric point of Mσ

gX ,nX. Let ℓ be a prime number prime to p. Then E. Ozman and R.

Pries proved the following result (cf. [OP, Theorem 1.1]):

Let ℓ = p be a prime number and nX = 0. Suppose that gX ≥ 2, and that0 ≤ σ ≤ gX with σ = 0 if gX = 2. Then every Galois admissible covering ofX• (i.e., Galois etale covering of X) with Galois group Z/ℓZ is new-ordinary.

By applying Ozman-Pries’ result mentioned above and similar arguments to the argumentsgiven in the proof of Theorem 3.6, we may obtained the following result:

Let ℓ = p be a prime number. Suppose that 0 ≤ σ ≤ gX with σ = 0 if gX = 2.Let D ∈ (Z/ℓZ)∼[DX ]

0. Let t ∈ N be a positive natural number such that

pt = 1 in (Z/ℓZ)×, n def= pt − 1, and m′ def

= n/ℓ. Write D′ for the divisorm′D ∈ (Z/nZ)∼[DX ]

0 when we identify Z/ℓZ with the unique subgroup of

Z/nZ of order ℓ. Let ([L], D) ∈ PX•,ℓ and

f • : Y • → X•

the Galois multi-admissible covering with Galois group Z/ℓZ induced by ([L], D).Suppose that X• be a pointed stable curve corresponding to a geometric pointover a generic point of Mσ

gX ,nX. Then f • is new-ordinary if and only if D′(j),

j ∈ {1, . . . , n}, is Frobenius-stable.

Remark 3.6.2. The author is interested in whether or not Theorem 3.6 can be generalizedto the case of arbitrary abelian admissible coverings. More precisely, we have the followingquestion:

Question: Suppose that X• is generic. Let f • : Y • → X• be a Galois multi-admissible covering whose Galois group is abelian. Can we find a necessaryand sufficient condition for the ordinariness of Y ?

4 Applications

In this section, we give some applications of Theorem 3.6. We maintain the notationintroduced in previous sections.

By applying Theorem 3.6, one may construct new-ordinary ramified coverings easilyfor generic curves. For instance, we have the following proposition.

Proposition 4.1. Let X• be a pointed stable curve of type (gX , nX) over an algebraicallyclosed field k of characteristic p > 0. Suppose that X• is generic. Then the followingstatements hold.

23

(i) Let m ∈ N be an arbitrary positive natural number prime to p. Suppose thatnX ≤ 1. Then there exists a new-ordinary Galois admissible covering

f • : Y • → X•

with Galois group Z/mZ such that f is etale.(ii) Let m ∈ N be an arbitrary positive natural number prime to p. Suppose that

nX ≥ 2 is an even number. We put DXdef= {x1, x2, . . . , x2d−1, x2d}. Let

Ddef=

d∑r=1

(arx2r−1 + brx2r)

such that 1 ≤ ar, br ≤ m− 1 and ar + br = m for each r ∈ {1, . . . , d}. Note that we have

D ∈ (Z/mZ)∼[DX ]0. Let ([L], D) ∈ PX•,m and

f • : Y • → X•

the Galois multi-admissible covering with Galois group Z/mZ induced by ([L], D). Thenf • is new-ordinary. In particular, there exists a new-ordinary Galois admissible coveringwhose Galois group is Z/mZ, and whose branch locus of f is equal to DX .

(iii) Let t be an arbitrary positive natural number, and ndef= pt − 1. Suppose that

p ≥ 5, and that nX ≥ 3 is an odd number. Then there exists a new-ordinary Galoismulti-admissible covering

f • : Y • → X•

with Galois group Z/nZ such that the branch locus of f is equal to DX .

Proof. (i) follows immediately from [N, Proposition 4]. Let us prove (ii). In order toverify (ii), we only need to prove that the restriction of f • on an arbitrary connectedcomponent of Y • is new-ordinary. Let t′ ∈ N be a positive natural number such thatpt

′= 1 in (Z/mZ)×. We put

D′ def= m′D =

d∑r=1

(m′arx2r−1 +m′brx2r) ∈ (Z/n′Z)∼[DX ]0

when we identify Z/mZ with the unique subgroup of Z/n′Z of order m, where n′ def= pt

′−1

and m′ def= n′/m. Note that we have n′ = m′ar +m′br for each r ∈ {1, . . . , r}. Moreover,

we see immediately that

deg(D′(j)(i)) = dn′, i ∈ {0, . . . , t′ − 1}, j ∈ {1, . . . , n′ − 1}.

This means that D′(j), j ∈ {1, . . . , n′}, is Frobenius-stable.Since L⊗n′ ∼= OX(−D)⊗m′ ∼= OX(−D′), we have ([L], D′) ∈ PX•,n′ . Let g• : Z• → X•

be the Galois multi-admissible covering over k with Galois group Z/n′Z correspondingto ([L], D′). Then Theorem 3.6 implies that g• is new-ordinary. Let W • be an arbitraryconnected component of Z• and

g•|W • : W • → X•

24

the Galois admissible covering over k induced by g•. Then we have g•|W • is new-ordinary.This completes the proof of (ii).

Next, let us prove (iii). Let nX = 2d + 1 and cdef= n/(p − 1). We put DX

def=

{x1, x2, . . . , x2d−1, x2d, x2d+1},

Qdef= cx2d−1 + cx2d + c(p− 3)x2d+1, Q′ def

=d−1∑r=1

(x2r−1 + (n− 1)x2r),

andD

def= Q+Q′.

Let j ∈ {1, . . . , n− 1}. We put αjdef= j − (p− 1)[j/(p− 1)] and βj

def= j(p− 3)− [j(p−

3)/(p−1)]. Note that we have that (p−1)|(2αj+βj), and that 2αj+βj ∈ {p−1, 2(p−1)}.Then we have

jQ = cjx2d−1 + cjx2d + cj(p− 3)x2d+1

= c(αjx2d−1+αjx2d+βjx2d+1)+(n[j/(p−1)]x2d−1+n[j/(p−1)]x2d+n[j(p−3)/(p−1)]x2d+1).

Thus, we have thatQ(j) = cαjx2d−1 + cαjx2d + cβjx2d+1.

Then we obtain that

deg(Q(j)(i)) = (2αj + βj)n/(p− 1), i ∈ {0, . . . , t− 1}.

Thus, we see immediately that Q(j), j ∈ {1, . . . , p− 1}, is Frobenius-stable.On the other hand, we see immediately that

deg(Q′(j)(i)) = dn, i ∈ {0, . . . , t− 1}, j ∈ {1, . . . , n− 1}.

This means that Q′(j), j ∈ {1, . . . , n− 1}, is Frobenius-stable. Thus, we have that D(j),

j ∈ {1, . . . , n−1}, is Frobenius-stable. Let ([L], D) ∈ PX•,n and f • : Y • → X• the Galoismulti-admissible covering over k with Galois group Z/nZ corresponding to ([L], D). ThenTheorem 3.6 implies that Y • is ordinary. This completes the proof of (ii).

Remark 4.1.1. Suppose that p ≤ 3, and that nX ≥ 3 is an odd number. Let f • : Y • →X• be a Galois admissible covering with Galois group Z/(p − 1)Z over k. Then we seeimmediately that the cardinality of the branch locus of f is ≤ nX − 1. Thus, (iii) doesnot hold when p ≤ 3.

Next, we consider an inverse Galois problem for X•. Let m be an arbitrary positive

natural number prime to p, n′ def= pt

′ − 1 such that m|n′, and m′ def= n′/m. Let G be a

finite group which is an extension of H = Z/mZ by a p-group P . Then we have

G ∼= P ⋊H.

Write Φ(P )def= P p[P, P ] for the Frattini subgroup of P and P

def= P/Φ(P ). We put

Gdef= G/Φ(P ). Then we obtain a Fp-representation

ρ : H → Aut(P ).

25

Let Z(H) be the set of irreducible characters of H with values in k and ζm a primitivemth root and χi, i ∈ Z/mZ, the irreducible character such that χi(1) = ζ im. Then wesee immediately that Z(H) = {χi}i∈Z/mZ. Let ρχi

: H → GL(P χi) be an irreducible

k-representation of H of character χi of degree 1. The canonical decomposition of P ⊗Fp kas a k[H]-module is given by

P ⊗Fp k =⊕

i∈Z/mZ

Pmχiχi

.

Then we have the following result.

Proposition 4.2. We maintain the notation introduced above. Let X• be a pointed sta-ble curve of type (gX , nX) over an algebraically closed field k of characteristic p > 0.Suppose that X• is generic. Let D ∈ (Z/mZ)∼[DX ]

0. Write D′ for the divisor m′D ∈(Z/nZ)∼[DX ]

0 when we identify Z/mZ with the unique subgroup of Z/n′Z of order m.

Let ([L], D) ∈ PX•,m and α ∈ Hom(ΠabX• ,Z/mZ) the element induced by ([L], D). We

putϕ : ΠX• ↠ Πab

X•α→ H = Z/mZ.

Suppose that D′(im′), i ∈ {1, . . . ,m−1}, is Frobenius-stable. Then an embedding problem(ϕ : ΠX• → H,G ↠ H) has a solution if and only if

mχi≤

{gX , if i = 0,gX + s(D′(im′))− 1, if i ∈ {1, . . . ,m− 1}.

Proof. The proposition follows immediately from Theorem 3.6 and [B, Proposition 2.4and Proposition 2.5].

26

References

[B] I. Bouw, The p-rank of ramified covers of curves, Compositio Math. 126(2001), 295-322.

[C] R. Crew, Etale p-covers in characteristic p, Compositio Math. 52 (1984),31–45.

[FJ] M. D. Fried, M. Jarden, Field arithmetic. Third edition. Ergebnisse der Math-ematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Math-ematics 11. Springer-Verlag, Berlin, 2008.

[G] A. Grothendieck, Revetements Etales et Groupe Fondamental (SGA1),Seminaire de Geometrie Algebrique de Bois-Marie 1960/61., Lecture Notesin Math. 224, Springer-Verlag, 1971.

[N] S. Nakajima, On generalized Hasse-Witt invariants of an algebraic curve,Galois groups and their representations (Nagoya 1981) (Y. Ihara, ed.), Adv.Stud. Pure Math, 2, North-Holland Publishing Company, Amsterdam, 1983,69–88.

[OP] E. Ozman, R. Pries, On the existence of ordinary and almost ordinary Prymvarieties, to appear in Asian Journal of Math.

[PaSt] A. Pacheco, K. Stevenson, Finite quotients of the algebraic fundamentalgroup of projective curves in positive characteristic. Pacific J. Math. 192(2000), 143–158.

[PoSa] F. Pop, M. Saıdi, On the specialization homomorphism of fundamentalgroups of curves in positive characteristic. Galois groups and fundamentalgroups, 107–118, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press,Cambridge, 2003.

[R1] M. Raynaud, Sections des fibres vectoriels sur une courbe. Bull. Soc. math.France 110 (1982), 103–125.

[R2] M. Raynaud, Sur le groupe fondamental d’une courbe complete en car-acteristique p > 0. Arithmetic fundamental groups and noncommutative al-gebra (Berkeley, CA, 1999), 335-351, Proc. Sympos. Pure Math., 70, Amer.Math. Soc., Providence, RI, 2002.

[Sa] A. Sarashina, Reconstruction of one-punctured elliptic curves in posi-tive characteristic by their geometric fundamental groups, to appear inManuscripta Math.

[Se] J-P. Serre, Sur la topologie des varietes algebriques en caracteristique p.Symp. Int. Top. Alg., Mexico (1958), 24–53.

[T1] A. Tamagawa, On the fundamental groups of curves over algebraically closedfields of characteristic > 0. Internat. Math. Res. Notices (1999), 853–873.

27

[T2] A. Tamagawa, Fundamental groups and geometry of curves in positivecharacteristic. Arithmetic fundamental groups and noncommutative algebra(Berkeley, CA, 1999), 297–333, Proc. Sympos. Pure Math., 70, Amer. Math.Soc., Providence, RI, 2002.

[T3] A. Tamagawa, On the tame fundamental groups of curves over algebraicallyclosed fields of characteristic > 0. Galois groups and fundamental groups,47–105, Math. Sci. Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge,2003.

[T4] A. Tamagawa, Finiteness of isomorphism classes of curves in positive charac-teristic with prescribed fundamental groups. J. Algebraic Geom. 13 (2004),675–724.

[V] I. Vidal, Contributions a la cohomologie etale des schemas et des log-schemas,These, U. Paris-Sud (2001).

[Y1] Y. Yang, On the admissible fundamental groups of curves over algebraicallyclosed fields of characteristic p > 0, Publ. Res. Inst. Math. Sci. 54 (2018),649–678.

[Y2] Y. Yang, Tame anabelian geometry and moduli spaces of curves over alge-braically closed fields of characteristic p > 0, RIMS Preprint 1879. See alsohttp://www.kurims.kyoto-u.ac.jp/∼yuyang/

[Y3] Y. Yang, The combinatorial mono-anabelian geometry of curves over alge-braically closed fields of positive characteristic I: combinatorial Grothendieckconjecture. See also http://www.kurims.kyoto-u.ac.jp/∼yuyang/

[Y4] Y. Yang, On the averages of generalized Hasse-Witt invariants of pointedstable curves in positive characteristic, to appear in Math. Z.

[Y5] Y. Yang, Moduli spaces of fundamental groups of curves in positive charac-teristic I, in preparation.

[Y6] Y. Yang, Moduli spaces of fundamental groups of curves in positive charac-teristic II, in preparation.

[Z] B. Zhang, Revetements etales abeliens de courbes generiques et ordinarite,Ann. Fac. Sci. Toulouse Math. (5) 6 (1992), 133–138.

Yu Yang

Address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

E-mail: yuyang@kurims.kyoto-u.ac.jp

28