Math. J. Okayama Univ. 64 (2022), 1–11

A NOTE ON TORSION POINTS ON AMPLE DIVISORSON ABELIAN VARIETIESYuichiro Hoshi
Abstract. In the present paper, we consider torsion points on ample
divisors on abelian varieties. We prove that, for each integer n ≥ 2, aneffective divisor of level n on an abelian variety does not contain thesubgroup of n-torsion points. Moreover, we also discuss an applicationof this result to the study of the p-rank of cyclic coverings of curves inpositive characteristic.

Introduction
In the present paper, we consider torsion points on ample divisors on
abelian varieties. The main result of the present paper is as follows [cf.
Corollary 1.8, (i)].
Theorem A. Let k be an algebraically closed field, A an abelian variety overk, D an effective divisor on A, and n ≥ 2 an integer invertible in k. Supposethat the effective divisor D is of level n, i.e., that there exists an effective
divisor D1 on A such that D1 gives rise to a principal polarization on A,
and, moreover, D is linearly equivalent to nD1 [cf. Definition 1.3, (ii);also Remark 1.3.1]. Then the subgroup of n-torsion points of A is notcontained in Supp(D).
Here, let us recall that R. Auffarth, G. P. Pirola, and R. S. Manni provedthat if D is an effective divisor on an abelian variety of dimension g ≥ 1
over the field of complex numbers that gives rise to a principal polarizationon the abelian variety, then, for each integer n ≥ 3, the set of n-torsion(respectively, 2-torsion) points on Supp(D) is of cardinality ≤ n2g −(g+1)ng(< n2g ) (respectively, ≤ 22g − 2g−1 g − 2g (< 22g )) [cf. [1], Theorem 1.1].
Theorem A may be regarded as a partial generalization of this result [cf.
Remark 1.8.1].
In §2 of the present paper, we apply Theorem A and Raynaud’s theory oftheta divisors [cf. [5]] to obtain an application to the study of the p-rank ofcyclic coverings of curves in positive characteristic. One consequence of ourapplication is as follows [cf. Theorem 2.7, (i)].
Theorem B. Let p be an odd prime number, k an algebraically closed fieldof characteristic p, and X a projective smooth connected curve over k of
Mathematics Subject Classification. Primary 14K12; Secondary 14H30.
Key words and phrases. abelian variety, torsion point, curve, p-rank.
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Y. HOSHI

genus ≥ 2. Then there exist a positive integer n such that p − 1 ∈ nZ and afinite étale cyclic covering of X of degree n whose Jacobian variety is of
positive p-rank.
Here, let us recall that M. Raynaud proved that, in the situation of Theorem B, the étale fundamental group of X is not pro-prime-to-p [cf. [5],Corollaire 4.3.2]. In §2 of the present paper, we also derive a refinement of
this result from Theorem B [cf. Remark 2.8.1, (iii)].
1. Torsion Points on Ample Divisors on Abelian Varieties
In the present §1, we discuss torsion points on ample divisors on abelianvarieties and prove the main result of the present paper [cf. Corollary 1.8
below]. In the present §1, let g be a positive integer, k an algebraicallyclosed field,A
an abelian variety over k of dimension g, n a positive integer, andL
an ample invertible sheaf on A of separable type [cf. [2], p.289].
Definition 1.1. We shall write A[n] ⊆ A for the closed subgroup schemeof A obtained by forming the kernel of the endomorphism of A given bymultiplication by n.
Lemma 1.2. The following four conditions are equivalent:(1) There exist an ample invertible sheaf L1 on A of degree one [cf.
∼[2], p.289, (III)] and an isomorphism L → L⊗n1 .
g(2) The invertible sheaf L is of degree n , and, moreover, there exist∼
an ample invertible sheaf L1 on A and an isomorphism L → L⊗n
1 .
(3) The equality H(L) = A[n] [cf. [2], p.288, Definition] holds, and,moreover, there exist an ample invertible sheaf L1 on A and an iso∼morphism L → L⊗n
1 .
(4) The equality H(L) = A[n] holds.
Proof. The equivalence (1) ⇔ (2) follows from [2], p.289, (II). Moreover, theequivalence (3) ⇔ (4) follows from [3], p.214, Theorem 3. Next, since thegroup scheme A[n] is of degree n2g over k [cf. [3], p.60, Proposition, (1)], theimplication (3) ⇒ (2) follows from [2], p.289, (IV).
Finally, we verify the implication (2) ⇒ (3). Suppose that condition (2)is satisfied. Then since L is isomorphic to L⊗n
1 [cf. condition (2)], the homo∧morphisms Λ(L), Λ(L1 ) : A → A [cf. [2], p.289, (IV)] satisfy the equality
Λ(L) = n·Λ(L1 ). Thus, it follows that A[n] ⊆ Ker(n·Λ(L1 )) = Ker(Λ(L)) =TORSION POINTS ON AMPLE DIVISORS

3

H(L). On the other hand, since L is of degree ng [cf. condition (2)], it follows from [2], p.289, (IV), that the group scheme H(L) is of degree n2g overk. Thus, since the group scheme A[n] is of degree n2g over k [cf. [3], p.60,
Proposition, (1)], the equality H(L) = A[n], hence also condition (3), holds,
as desired. This completes the proof of the implication (2) ⇒ (3), hence
also of Lemma 1.2.
Definition 1.3.
(i) We shall say that the ample invertible sheaf L of separable type isof level n if L satisfies the four conditions [i.e., with respect to the
fixed “n”] in the statement of Lemma 1.2.
(ii) We shall say that an effective divisor D on A is of level n if theinvertible sheaf OA (D) is [ample, of separable type, and] of level n.
Remark 1.3.1. Let M be an invertible sheaf on A. Then it is immediatethat M gives rise to a principal polarization on A if and only if M is [ample,of separable type, and] of level one.
Lemma 1.4. Let M be an invertible sheaf on A algebraically equivalent
to L. Then the following hold:(i) There exists a closed point a ∈ A of A such that L is isomorphic
to Ta∗ M [cf. [2], p.288, Definition].
(ii) The invertible sheaf M is ample and of separable type.
(iii) Suppose that M is of level n [cf. (ii)]. Then L is of level n.
Proof. First, we verify assertion (i). Let us first observe that the homomorphism A(k) → Pic0 (A) determined by Λ(L) is surjective [cf. [2], p.289,(IV)]. Thus, there exists a closed point a ∈ A of A such that M ⊗OA L−1∗ L⊗−1
is isomorphic to T−a
OA L . Thus, we conclude that L is isomorphic to
Ta∗ M, as desired. This completes the proof of assertion (i). Assertions (ii),(iii) follow from assertion (i). This completes the proof of Lemma 1.4. 
Lemma 1.5. Suppose that L is of level n. Then the following two conditions are equivalent:(1) The inequality n > 1 holds.
(2) The invertible sheaf L is generated by global sections.
Proof. The implication (1) ⇒ (2) follows immediately from [3], pp.57-58,
Application 1, (iii). Next, to verify the implication (2) ⇒ (1), assume thatcondition (2) is satisfied, but that condition (1) is not satisfied [i.e., thatn = 1]. Then it follows from [2], p.289, (II), that Γ(A, L) is of dimension
one. Thus, since L is generated by global sections [cf. condition (2)], theinvertible sheaf L is trivial. In particular, since [we have assumed that]
L is ample, we conclude that g = 0. Thus, since [we have assumed that]

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g > 0, we obtain a contradiction, as desired. This completes the proof of
the implication (2) ⇒ (1), hence also of Lemma 1.5.
One main technical observation of the present paper is as follows.
Lemma 1.6. Let D be an effective divisor on A obtained by forming thezero locus of a nonzero global section of the invertible sheaf L. Write
H(D) ⊆ H(L) for the subgroup of H(L) consisting of a ∈ A such thatTa∗ D = D. Let H ⊆ H(L) be a subgroup of H(L) such that H + H(D)def

( = { h + hd ∈ H(L) | h ∈ H, hd ∈ H(D) }) = H(L). Suppose that theinclusion
H ⊆ Supp(D)holds. Then the subset H ⊆ A of A is contained in the base locus of the[complete linear system associated to the] invertible sheaf L.
Proof. Let s ∈ Γ(A, L) be a nonzero global section of L whose zero locus isgiven by D.
Here, let us recall the exact sequence0/ k×/ G(L)/ H(L)/0in [2], p.290, concerning the theta group G(L) associated to L. It follows
from the definition of G(L) that there exists a natural action of G(L) on
the linear space Γ(A, L) over k, which restricts to the natural action of thesubgroup k × ⊆ G(L) on Γ(A, L) [cf. [2], p.295, Definition]. In particular,(a) for each a ∈ H(L), if ea ∈ G(L) is a lifting of a ∈ H(L), then the zero
∗ D.
locus of the nonzero global section e
a · s ∈ Γ(A, L) is given by T−a
Now let us fix a subsete ⊆ G(L)H

e ֒→ G(L) ։ H(L) determines a bijectionof G(L) such that the composite H∼
e → H. Then since [we have assumed that] the inclusion H ⊆ Supp(D)H
holds, it follows from (a) that,∗ H” of A —e the subset H ⊆ A [i.e., the subset “T−a(b) for every ea ∈ H,
e in H] is contained in thewhere we write a for the image of ea∈Hzero locus of the nonzero global section e
a · s ∈ Γ(A, L).
Next, let us observe that it follows immediately from (a), together with
our assumption that H + H(D) = H(L), that(c) the linear subspace of Γ(A, L) generated by the G(L)-orbit of s ∈Γ(A, L) coincides with the linear subspace of Γ(A, L) generated by
the subset {ea · s}ea∈He ⊆ Γ(A, L).

TORSION POINTS ON AMPLE DIVISORS

5

On the other hand, it follows from [2], p.297, Theorem 2, that the action ofG(L) on Γ(A, L) is irreducible. Thus, we conclude from (c) that(d) the subset {ea · s}ea∈He ⊆ Γ(A, L) generates the linear space Γ(A, L).
Thus, it follows from (b) and (d) that the subset H ⊆ A is contained in
the base locus of the invertible sheaf L, as desired. This completes the proof
of Lemma 1.6.
Theorem 1.7. Let k be an algebraically closed field, A an abelian variety
over k, and D an effective divisor on A. Suppose that the invertible sheaf
OA (D) is ample, of separable type [cf. [2], p.289], and generated by
global sections. Then the following hold:(i) Recall the closed subgroup scheme H(OA (D)) ⊆ A of A defined in [2],p.288, Definition. Then H(OA (D)) is not contained in Supp(D).
(ii) Write deg(D) for the degree of the ample invertible sheaf OA (D)[cf. [2], p.289, (III)]. Then A[deg(D)] [cf. Definition 1.1] is notcontained in Supp(D).
Proof. Assertion (i) follows from Lemma 1.6. Assertion (ii) follows from
assertion (i), together with the inclusion H(OA (D)) ⊆ A[deg(D)] [cf. [2],p.289, (IV); [2], p.293, Theorem 1; also the first Definition in [2], p.294]. 
The main result of the present paper is as follows.
Corollary 1.8. Let k be an algebraically closed field, A an abelian variety
over k, D an effective divisor on A, and n a positive integer invertible in k.
Suppose that the effective divisor D is of level n [cf. Definition 1.3, (ii)].
Then the following hold:(i) Suppose that n ≥ 2. Then A[n] is not contained in Supp(D).
(ii) Suppose that n = 1. Then, for each integer m ≥ 2 invertible in k,
A[m] is not contained in Supp(D).
Proof. Let us recall from condition (4) of Lemma 1.2 that the equality
H(OA (D)) = A[n] holds. Thus, assertion (i) follows from Lemma 1.5 andTheorem 1.7, (i).
Next, we verify assertion (ii). Let m ≥ 2 be an integer invertible in k.
Then since D is of level one, it is immediate that mD is of level m. Thus,
since Supp(mD) = Supp(D), it follows from assertion (i) that A[m] is notcontained in Supp(D), as desired. This completes the proof of assertion (ii),hence also of Corollary 1.8.

Remark 1.8.1. R. Auffarth, G. P. Pirola, and R. S. Manni proved that, inthe situation of Corollary 1.8, if, moreover, k is the field of complex numbers,
and n = 1 [i.e., the divisor D gives rise to a principal polarization on A —cf. Remark 1.3.1], then, for each integer m ≥ 3, the set A[m] ∩ Supp(D)6

Y. HOSHI(respectively, A[2] ∩ Supp(D)) is of cardinality ≤ n2g − (g + 1)ng (< n2g )(respectively, ≤ 22g − 2g−1 g − 2g (< 22g )) — where we write g for thedimension of A [cf. [1], Theorem 1.1]. Corollary 1.8 may be regarded as apartial generalization of this result.
2. Application: p-Rank of Cyclic Coverings of Curves
In the present §2, we apply the main result of the present paper andRaynaud’s theory of theta divisors [cf. [5]] to obtain an application to thestudy of the p-rank of cyclic coverings of curves in positive characteristic[cf. Theorem 2.7 below]. In the present §2, let p be a prime number, k an
algebraically closed field of characteristic p, g ≥ 2 an integer,Xa projective smooth connected curve over k of genus g, n ≥ 2 an integer
invertible in k, andL
an invertible sheaf on X of order n.
Definition 2.1. We shall write X F for the projective smooth connectedcurve over k obtained by forming the base-change of X by the absolute
Frobenius endomorphism of k, LF for the invertible sheaf on X F obtainedby forming the base-change of L by the absolute Frobenius endomorphismof k, and Φ : X → X F for the relative Frobenius morphism associated to X
over k.
Remark 2.1.1. Let us recall that we have a natural isomorphism of invertible sheaves on X
∼ / ∗ FΦ L
L⊗pgiven by, for each local section l of L, mapping l⊗p to Φ−1 lF — where wewrite lF for the local section of LF obtained by forming the base-change ofthe local section l by the absolute Frobenius endomorphism of k. Let us
identify L⊗p with Φ∗ LF by means of this isomorphism.
Definition 2.2.
(i) Let i be an element of {1, . . . , n}. Then we shall write
γL,i : H 1 (X F , (LF )⊗i )/ H 1 (X, L⊗pi )for the k-linear homomorphism obtained by applying “H 1 (X F , (−)⊗OX F(LF )⊗i )” to the homomorphism OX F → Φ∗ OX determined by Φ [cf.
also Remark 2.1.1].

TORSION POINTS ON AMPLE DIVISORS

7(ii) We shall say that the invertible sheaf L is new-ordinary if, for every
element i ∈ {1, . . . , n − 1} with nZ + iZ = Z, the homomorphism
γL,i of (i) is an isomorphism.
Remark 2.2.1.
(i) One verifies immediately from the theory of finite étale cyclic coverings and generalized Hasse-Witt invariants [cf., e.g., [6], §2.1, or [7],pp.73-74] thatthe existence of a new-ordinary invertible sheaf on X oforder nis equivalent to
the existence of a new-ordinary finite étale cyclic coveringof X of degree n, i.e., a finite étale cyclic covering of X ofdegree n that has a new ordinary part in the sense of [6],Définition 2.1.1,
which thus implies
the existence of a finite étale cyclic covering of X of degree
n whose Jacobian variety is of p-rank ≥ (g − 1) · ♯(Z/nZ)×(> 0).
(ii) Suppose that p − 1 ∈ nZ. Then each trivialization ι of L⊗n determines an isomorphism of invertible sheaves on X
ι(p−1)/n : L⊗p

∼

/ L.

Thus, the homomorphism γL,i may be “identified”, i.e., by means ofι(p−1)/n , with the homomorphism
H 1 (X F , (LF )⊗i )/ H 1 (X, L⊗i ).

In particular, one verifies immediately from the theory of finite étale
cyclic coverings and generalized Hasse-Witt invariants [cf., e.g., [6],§2.1, or [7], pp.73-74] thatthe existence of an invertible sheaf M on X of order n suchthat the homomorphism γM,i is an isomorphism for somei ∈ {1, . . . , n − 1}
implies
the existence of a finite étale cyclic covering of X of degree
n whose Jacobian variety is of p-rank ≥ dimk H 1 (X, L⊗i ) =g − 1 (> 0).
In the remainder of the present §2, write J F for the Jacobian variety of
X F and B F for the OX F -module obtained by forming the cokernel of thehomomorphism OX F → Φ∗ OX determined by Φ. Moreover, let us fix auniversal invertible sheaf P F on X F ×k J F of degree zero.

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Y. HOSHI

Definition 2.3. We shall writeΘB F ⊆ J F
for the closed subscheme of J F defined by the zeroth Fitting ideal of thecoherent OJ F -moduleprpr
R1 (X F ×k J F →2 J F )∗ P F ⊗OX F × J F (X F ×k J F →1 X F )∗ B F
k[cf. also [7], Remark 1.1].

Proposition 2.4. The following hold:(i) The closed subscheme ΘBF ⊆ J F of J F forms a [necessarily effective]
divisor on J F of level p − 1 [cf. Definition 1.3, (ii)].
(ii) Let x ∈ J F be a closed point of J F and MF an invertible sheaf on
X F of degree zero whose isomorphism class corresponds to x ∈ J F .
Then the following three conditions are equivalent:(1) The closed point x ∈ J F is not contained in ΘBF .
(2) The equality Γ(X F , MF ⊗OX F B F ) = {0} holds.
(3) The equality H 1 (X F , MF ⊗OX F B F ) = {0} holds.
(iii) The underlying closed subset of the closed subscheme ΘBF ⊆ J F of
J F is stabilized by the automorphism of J F given by multiplication
by −1.
Proof. First, we verify assertion (i). It follows from [5], Théorème 4.1.1,
that the closed subscheme ΘBF ⊆ J F of J F forms a [necessarily effective]
divisor on J F . Moreover, since [it is well-known that] the “classical theta
divisor” on J F gives rise to a principal polarization on J F , it follows from[5], Proposition 1.8.1, (2) [cf. also [5], §4], together with Lemma 1.4, (iii),of the present paper [cf. also Remark 1.3.1 of the present paper], that thedivisor determined by ΘBF ⊆ J F is of level p − 1, as desired. This completes
the proof of assertion (i).
Assertion (ii) follows immediately from the definition of the closed subscheme ΘBF ⊆ J F [cf. also [5], §4; [7], Lemma 1.2]. Finally, we verify
assertion (iii). Let us recall from the discussion preceding [5], Théorème∼
4.1.1, that there exists an isomorphism B F → HomOX F (B F , Ω1X F /k ) of OX F modules. Thus, assertion (iii) follows immediately from assertion (ii), together with Serre duality. This completes the proof of assertion (iii), hence
also of Proposition 2.4.
Lemma 2.5. The following hold:(i) Suppose that p 6= 2. Then J F [p − 1] [cf. Definition 1.1] is not contained in ΘBF .

TORSION POINTS ON AMPLE DIVISORS

9(ii) Suppose that p = 2. Then, for each odd integer m ≥ 3, J F [m] is
not contained in ΘBF .
Proof. These assertions follow from Corollary 1.8 and Proposition 2.4, (i).

Lemma 2.6. The following hold:(i) Let i be an element of {1, . . . , n}. Then it holds that the homomorphism γL,i is an isomorphism if and only if the closed point of J F
that corresponds to (LF )⊗i is not contained in ΘBF ⊆ J F .
(ii) It holds that the Jacobian variety of X is ordinary if and only if
the identity element of J F is not contained in ΘBF ⊆ J F .
(iii) It holds that the invertible sheaf L is new-ordinary if and only if,for every element i ∈ {1, . . . , n−1} with nZ+iZ = Z, the closed pointof J F that corresponds to (LF )⊗i is not contained in ΘBF ⊆ J F .
(iv) Suppose that n ∈ {2, 3, 4, 6}. Then it holds that the invertible sheaf L
is new-ordinary if and only if there exists an element i ∈ {1, . . . , n−1} such that nZ + iZ = Z, and, moreover, the closed point of J F that
corresponds to (LF )⊗i is not contained in ΘBF ⊆ J F .
Proof. Assertion (i) follows immediately from Proposition 2.4, (ii), together
with the definition of the OX F -module B F . Assertions (ii), (iii) follow from
assertion (i) [cf. also [6], §2.1]. Finally, we verify assertion (iv). The necessity
follows from assertion (iii). The sufficiency follows from Proposition 2.4, (iii),
and assertion (iii). This completes the proof of assertion (iv), hence also ofLemma 2.6.

One interesting application of the main result of the present paper is asfollows.
Theorem 2.7. Let p be a prime number, k an algebraically closed field of
characteristic p, and X a projective smooth connected curve over k of genus≥ 2. Then the following hold:(i) Suppose that p 6= 2. Then there exist a positive integer n such thatp − 1 ∈ nZ and a finite étale cyclic covering of X of degree n
whose Jacobian variety is of positive p-rank.
(ii) Suppose that the Jacobian variety of X is not ordinary. Let n
be an integer such that (p, n) ∈ {(2, 3), (3, 2)}. Then there exists a
new-ordinary finite étale cyclic covering of X of degree n, i.e.,a finite étale cyclic covering of X of degree n that has a new
ordinary part in the sense of [6], Définition 2.1.1.
Proof. Assertion (i) follows immediately — in light of Remark 2.2.1, (ii)— from Lemma 2.5, (i), and Lemma 2.6, (i), (ii). Assertion (ii) follows

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Y. HOSHI

immediately — in light of Remark 2.2.1, (i) — from Lemma 2.5, (i), (ii),and Lemma 2.6, (ii), (iv).
Remark 2.7.1. Some results closely related to the content of Theorem 2.7
are as follows: In the situation of Theorem 2.7, suppose that X is of genus
g (≥ 2). Then:(i) M. Raynaud proved that if, moreover, l is a prime number such thatl + 1 ≥ (p − 1)3g−1 g!, then there exists a new-ordinary finite étale
cyclic covering of X of degree l [cf. [5], Théorème 4.3.1; also [7],Remark 3.11].
(ii) S. Nakajima proved that if, moreover, (g, p) = (2, 2), and the Jacobian variety of X is not ordinary [i.e., the curve X is either of typeI or of type II in the sense of [4], §6], then every finite étale cyclic
covering of X of degree three is new-ordinary [i.e., the curve X is
3-ordinary in the sense of the discussion at the beginning of [4], §4][cf. [4], §6].
Corollary 2.8. Let p be a prime number, k an algebraically closed fieldof characteristic p, and X a projective smooth connected curve over k ofgenus ≥ 2. Write π1 (X) for the étale fundamental group [for some choiceof basepoint] of X,p − 1 if p 6= 2
def
np =
3
if p = 2,
N ⊆ π1 (X) for the normal open subgroup of π1 (X) obtained by forming thekernel of the natural surjective homomorphism
π1 (X)/ / π1 (X)ab ⊗ b (Z/np Z),
Z

and Y → X for the finite étale abelian covering that corresponds to thenormal open subgroup N ⊆ π1 (X). Then the Jacobian variety of Y is of
positive p-rank. In particular, the maximal pro-p abelian quotient of
N is nontrivial [cf. Remark 2.8.1, (i), below].
Proof. This assertion is a formal consequence of Theorem 2.7, (i), (ii).

Remark 2.8.1.
(i) Let us recall that it is well-known that, in the situation of Corollary 2.8, the maximal pro-p abelian quotient of π1 (X) has a natural
structure of finitely generated free Zp -module whose rank coincides
with the p-rank of the Jacobian variety of X.
(ii) Let G be a profinite group and l a prime number. Then it is immediate that the following three conditions are equivalent:(1) The profinite group G is pro-prime-to-l.

TORSION POINTS ON AMPLE DIVISORS

11(2) The maximal pro-l abelian quotient of every open subgroup of
G is trivial.
(3) An arbitrary [or, alternatively, some] pro-l Sylow subgroup of
G is trivial.
(iii) M. Raynaud proved that, in the situation of Corollary 2.8, the profinite group π1 (X) is not pro-prime-to-p [cf. [5], Corollaire 4.3.2]. Let
us observe that it follows from the observation of (ii) that Corollary 2.8 may be regarded as a refinement of this result by Raynaud.
Acknowledgement
This research was supported by JSPS KAKENHI Grant Number 18K03239
and by the Research Institute for Mathematical Sciences, an International
Joint Usage/Research Center located in Kyoto University.
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Amer. Math. Soc. 145 (2017), no. 1, 89-99.
[2] D. Mumford, On the equations defining abelian varieties. I. Invent. Math. 1, 1966,
287-354.
[3] D. Mumford, Abelian varieties. With appendices by C. P. Ramanujam and Yuri
Manin. Corrected reprint of the second (1974) edition. Tata Institute of Fundamental
Research Studies in Mathematics, 5. Published for the Tata Institute of Fundamental
Research, Bombay; by Hindustan Book Agency, New Delhi, 2008.
[4] S. Nakajima, On generalized Hasse-Witt invariants of an algebraic curve. Galois
groups and their representations (Nagoya, 1981), 69-88, Adv. Stud. Pure Math., 2,
North-Holland, Amsterdam, 1983.
[5] M. Raynaud, Sections des fibrés vectoriels sur une courbe. Bull. Soc. Math. France
110 (1982), no. 1, 103-125.
[6] M. Raynaud, Sur le groupe fondamental d’une courbe complète en caractéristiquep > 0. Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA,1999), 335-351, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence, RI,
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[7] A. Tamagawa, On the tame fundamental groups of curves over algebraically closed
fields of characteristic > 0. Galois groups and fundamental groups, 47-105, Math. Sci.
Res. Inst. Publ., 41, Cambridge Univ. Press, Cambridge, 2003.
Yuichiro Hoshi
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502 Japane-mail address: yuichiro@kurims.kyoto-u.ac.jp(Received November 20, 2019 )(Accepted March 16, 2021 )