UTMS 2002–8

February 12, 2002

Abelian étale coverings of curves over
local fields and its application to modular curves
by
Teruyoshi Yoshida

T

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANABELIAN ÉTALE COVERINGS OF CURVES OVER LOCALFIELDS AND ITS APPLICATION TO MODULAR CURVESTERUYOSHI YOSHIDA

Abstract. We give a geometric bound of a part of the abelian étale fundamental group of curves over local fields, in particular in the case of bad
reduction. We apply the result to the modular curve X0 (p)/Qp to show thatthe unramified abelian covering between X1 (p) → X0 (p) (Shimura covering)uses up all the possible ramification over the special fiber.

Contents
1. Introduction

1

2. Preliminaries on abelian étale fundamental groups

3

3. The prime-to-p part of π1ab (X)georam

6

4. The p-primary part for low absolute ramiﬁcation case8

5. The case of the modular curve X0 (p)10References

12

1. Introduction
The problem of classifying the abelian étale coverings of a proper smooth curveis the central problem of the geometric class ﬁeld theory, and there is an arithmetical interest when the base ﬁeld is a number ﬁeld or a local ﬁeld. There is awell-formulated class ﬁeld theory for these curves (e.g. Kato-Saito[6], Saito[14])but there are very few examples where the abelian étale coverings are explicitlyclassiﬁed. The most remarkable known example is the case of modular curve
X0 (p) over Q (Mazur[8], Introduction, Theorem (2)), which makes use of the
deep theory of integral Hecke algebras. We give the corresponding result forX0 (p) over Qp , by proving general results for the curves over local ﬁelds.
For a proper variety X over a ﬁeld K, let π1ab (X) be the abelian étale fundaab
mental group of X. There is a natural surjection π1ab (X) → Gab
K where GK =Date: February 13, 2002.
1

2

TERUYOSHI YOSHIDA

Gal(Kab /K) is the Galois group of the maximal abelian extension of K, and wedenote the kernel by π1ab (X)geo :/ π ab (X)geo

0/ π ab (X)

1

/ Gab

1

/0

K

For a proper smooth geometrically irreducible variety X over a local field K, i.e.
a complete discrete valuation ﬁeld K with ﬁnite residue ﬁeld F with charF = p,
 r bywe showed in our previous paper [15] that π1ab (X)geo is an extension of Zgeo
aba ﬁnite torsion group π1 (X)tor . Here the rank r is the F -rank (the dimensionof the maximal F -split subtorus) of the special ﬁber of the Néron model of the
Albanese variety of X.
In this paper, we conﬁne ourselves in the case of curves, and investigate theabgeomost mysterious part of π1ab (X)geo
tor , namely π1 (X)ram which classiﬁes the abelianétale coverings of the generic ﬁber which are “completely ramiﬁed over the special
ﬁber”, deﬁned as follows. Let XF be the special ﬁber of the proper ﬂat regularmodel X over the integer ring OK (Abyhankar[1]). Then we have a natural map
π1ab (X)geo −→ π1ab (XF )geo which is surjective if X has a K-rational point, anddenote the kernel by π1ab (X)geo
ram :0/ π ab (X)geo
1ram/ π ab (X)geo/ π ab (X

1

1

F)

geo

/0

Then our main result is :Theorem 1.1. Assume that X admits a K-rational point.
(i) (Theorem 3.2) The dual of prime-to-p part of the finite abelian group π1ab (X)geo
raminjects to the component group Φ of the Néron model of the jacobian varietyof X.
(ii) (Theorem 4.1) Assume moreover that the absolute ramification index e of
K is less than p − 1, charK = 0, and X has semistable reduction over OK .
Then the p-primary part of π1ab (X)geo
ram vanishes.
Note that if XF is smooth, i.e. X has good reduction, π1ab (X)geo → π1ab (XF )geo
is isomorphic on the prime-to-p part by the proper smooth base change theorem
1
on Het. As π1ab (XF )geo can be calculated in principle from the special ﬁber XF ,we have control on the whole of π1ab (X) in the case e < p − 1, charK = 0 and X
has semistable reduction, in terms of the special ﬁber of the Néron model of thejacobian of X.
In the latter part of this paper, we give an application of these results tothe modular curve X0 (p), where X0 (p) is the usual modular curve classifying theelliptic curves with Γ0 (p)-structure (p is a prime). For the curve X0 (p)/Qp , above
theorem enables us to compute the group π1ab (X0 (p)/Qp )geo completely :Theorem 1.2. π1ab (X0 (p)/Qp )geo has the following structure :0/ Φ(J0 (p))/ π ab (X (p)/Q )geo
1

0

p

/ r

Z

/0

CURVES OVER LOCAL FIELDS

3

geo
ab
where Φ(J0 (p)) ∼= π1ab (X0 (p)/Qp )geo
ram = π1 (X0 (p)/Qp )tor is a cyclic group of orderequal to the numerator of p−1, and r = g+h−1
where g is the genus of X0 (p) and122
h is the number of the supersingular points defined over Fp .

Acknowledgements. This work is a part of the author’s master thesis at
University of Tokyo. The author would like to express his sincere gratitude to
his thesis adviser K. Kato for suggesting the problem and his constant encouragement.
Notations. Throughout this paper, a local field K means a complete discrete
valuation ﬁeld with ﬁnite residue ﬁeld F , with charF = p. OK is the integer ringof K. For any ﬁeld K, K is a separable closure of K, and GK = Gal(K/K) is theabsolute Galois group of K. For a variety X over any ﬁeld K, X = X ×K Spec(K).
For an abelian group X, its Pontrjagin dual is denoted by X ∨ = Hom(X, Q/Z).
∼Z= Zp is the proﬁnite completion of Z.
p

For a scheme X over OK , we denote the generic ﬁber and the special ﬁber of
X respectively by XK , XF . For any group scheme X, X[m] denotes the kernel ofthe multiplication-by-m map, which is also a group scheme. We use the notationµN = Gm [N ] for the group scheme of N -th roots of unity over an arbitrary base,without specifying the base scheme.
2. Preliminaries on abelian étale fundamental groups
Here we review some generalities on abelian étale fundamental groups, and ﬁx
the notations.
2.1. Geometric abelian étale fundamental groups. For any noetherian connected scheme X, the étale fundamental group π1 (X) is a proﬁnite group classifying ﬁnite étale coverings of X ([SGA1]), and we denote the maximal abelianquotient by π1ab (X). When X is a proper variety over a ﬁeld K, there is an exact
sequence (ibid., Exposé IX, Th. 6.1) :
(2.1)

1

/ π (X)1/ π1 (X)/ GK/1where K is a separable closure of K, X = X ×K Spec(K), and GK = Gal(K/K)
is the absolute Galois group of K. Therefore we have surjection π1ab (X) → Gab
K
where Gab
K = Gal(Kab /K) is the Galois group of the maximal abelian extensionof K, and denote the kernel by π1ab (X)geo :
(2.2)

0

/ π ab (X)geo
1/ π ab (X)
1

/ Gab
K

/0

Remark 2.1. When X has a K-rational point x, (2.1) and consequently also(2.2) has a splitting, and π1ab (X)GK ∼= π1ab (X)geo where the former group is the
coinvariant with respect to the GK -action by inner automorphisms. In this case,

4

TERUYOSHI YOSHIDA

π1ab (X)geo has a geometric interpretation as the group classifying the abelian ﬁnite
étale coverings of X in which x splits completely.
Now assume that K is a local field with the ﬁnite residue ﬁeld F with charF = p,
and let X be a proper ﬂat model of X over the integer ring OK , i.e. a proper ﬂat
scheme X → Spec OK with X ×OK Spec K ∼= X, and let XF = X ×OK Spec F
be the special ﬁber. Then we have a canonical surjection :π ab (X) −→ π ab (X ) ∼= π ab (XF )1

1

1

Here the latter isomorphism follows by [SGA1] Exposé X, Th. 2.1 (a part of1). We have the corresponding map on theproper base change theorem for Het
geometric part by the commutative diagram :
(2.3)

0

/ π ab (X)geo/ π ab (X)

/ Gab
K

/0

0


/ π ab (X/ π ab (X


/ GF

/0

1

1

1

geo
F)

1

F)

Definition 2.2. Denote the kernel of left and middle vertical maps of (2.3) by
π1ab (X)ram , π1ab (X)geo
ram respectively. (These groups classify the abelian étale coverings of X which become completely ramiﬁed over a part of XF when extended
to the coverings over XF .)By deﬁnition, we have an exact sequence :
(2.4)

0

/ π ab (X)geo
1/ π ab (X)georam

1/ π ab (X
1

F)

geo

1
As π1ab (X) is the Pontrjagin dual of the étale cohomology group Het(X, Q/Z),
we can pass to the dual and translate the above deﬁnition into the language of1
Het. By the Hochschild-Serre spectral sequece, we have a short exact sequence :0/ H 1 (GK , Q/Z)/ H 1 (X, Q/Z)et

/ H 1 (X, Q/Z)GK
et/0

where (−)GK is the Galois invariant. (Here we used H 2 (GK , Q/Z) = 0 for local
ﬁelds K, and the same holds when we replace K by the ﬁnite ﬁeld F .) We canview this exact sequence as the Pontrjagin dual of (2.2), and therefore we have acanonical isomorphism :(π ab (X)geo )∨ ∼(2.5)
= H 1 (X, Q/Z)GK1

et

Hence by taking the Pontrjagin dual of the exact sequence (2.4), we have :Lemma 2.3. Let X be a proper geometrically irreducible variety over a local fieldK, and let X be its proper flat model over OK , and denote the special fiber by
XF . Then there is a canonical exact sequence of abelian groups :(2.6)

1(XF , Q/Z)GF
Het/ H 1 (X, Q/Z)GKet/ (π ab (X)geo )∨
1

ram

/0

CURVES OVER LOCAL FIELDS

5

Remark 2.4. Note that when X has a K-rational point, it gives a splitting of
the exact sequence (2.2), and consequently all the vertical arrows of (2.3) aresurjective, and exact sequences (2.4) and (2.6) turn out to be the short exact
sequences.
2.2. Étale cohomology groups. Here we also review some properties of the1group Het
(X, Q/Z)GK for a proper geometrically irreducible curve X over a ﬁeld
K.
First recall that, for “prime to p” part, we have :Lemma 2.5 ([SGA4], Exposé IX, Corollaire 4.7). Let  = charK be a prime.
1
For any proper geometrically connected curve X over a field K, Het(X, Q /Z ) ∼=0∞
Pic (X)[ ](−1) as GK -modules.
More generally, considering the long exact sequence induced from the short
exact sequence :0/ 1 Z/Z
N

/ Q/Z

N

/ Q/Z

/0

1
1
1
(X, N1 Z/Z) = Het(X, Q/Z)[N ]. As Q/Z = lim
Z/Z and the étale
we have Het−→ N
N

cohomology commutes with the directed inductive limits ([SGA4] Exposé VII,
Théorème 5.7), we deduce that :

1
1
1
1
1
Z/Z)
=
Z/Z)
(X, Q/Z) = lim
H
(X,
H
(X,
Het
et
et
−→
N
N
N
N
1
(In particular, Het(X, Q/Z) is an torsion module.) Therefore we have :


1
1
1
1
(2.7)
(X, Q/Z)GK =Het(X, Q/Z)[N ]GK =Het(X, Z/Z)GK
Het
N
N
N
1
1
(X, Q/Z)GK to that of Het(X, Z/N Z)GK ,and can reduce the calculation of Het
for which we can make use of the following generalization of Lemma 2.5 :Lemma 2.6 (Milne[10], Theorem 3.9, Proposition 4.16). For any proper geometrically irreducible curve over a field K and integer N ≥ 1, we have a canonicalisomorphism of GK -modules :1
Het(X, Z/N Z) ∼= Homgp (µN , Pic0 (X)[N ])where Homgp denotes the GK -module consisting of homomorphisms as groupschemes. In particular, we have :1
Het(X, Z/N Z)GK ∼= HomK−gp (µN , Pic0 (X)[N ])where HomK−gp denotes the abelian group consisting of homomorphisms definedover K.

6

TERUYOSHI YOSHIDA

1Remark 2.7. This interpretation shows that, (π1ab (X)geo )∨ ∼(X, Q/Z)GK is= Het
0
the maximal µ-type subgroup of Pic (X) in the terminology of Mazur[8], I-3. This
re-interpretes the Mazur’s theory of the µ-type subgroup of the jacobian of themodular curves as the theory of π1ab (X)geo of the modular curves.

3. The prime-to-p part of π1ab (X)geo
ram
From this section, we assume that X is a proper smooth geometrically irreducible curve over a local field K.
In this case, the proper ﬂat regular model X always exists by Abyhankar[1],and denote its special ﬁber by XF . Let J = Pic0 (X) be the jacobian variety ofX, and denote the Néron model and its special ﬁber respectively by J and JF .
The quotient of JF by its connected component of the identity JF0 is a ﬁnite étalegroup Φ(J) over F , which we call the group of components of J (or of J, by anabuse of language).
Now we assume that X admits a K-rational point (see Remark 2.4), and ourstarting point of the investigation is the following :Lemma 3.1 (Raynaud[11], [SGA7] Exposé IX, (12.1.12)). Assume that X admits a K-rational point. Then we have Pic0 (XF ) ∼= JF0 as smooth group schemes
over F .
In this section we treat the prime-to-p part of π1ab (X)geo
ram , and the main result
is stated as follows :Theorem 3.2. Assume that X admits a K-rational point. Then there is an∨injection of finite abelian groups (π1ab (X)geo
ram )not p −→ Φ(J), where (−)not p denotes
the prime-to-p part (p = charF ).
In this section, we ﬁx some notations on Galois modules, and give the proof of
the above theorem. As the theorem can be proved by establishing the injectivity
on -primary parts for each prime number  = p = charF , we ﬁx an  = p in theremainder of this section.
3.1. Galois modules. For any smooth group scheme X of ﬁnite type over K,
deﬁne pro- (resp. ind-) group scheme T (X) (resp. X[∞ ]) by :X[n ], X[∞ ] = lim
X[n ]T (X) = lim
←−
−→
n

n

where X[m] denotes the kernel of the multiplication-by-m map which is an étalegroup scheme over K, and inductive (resp. projective) limit is taken with respect
to the canonical inclusions (resp. multiplication-by- maps). We use the same
notations for group schemes over OK or F . We often identify T (X), X[∞ ] withthe associated Galois modules, which are Z -modules and satisﬁes :X[∞ ] ∼= T (X) ⊗ (Q /Z ), T (X) = Hom(Q /Z , X[∞ ])CURVES OVER LOCAL FIELDS

7

Now we ﬁx the notations concerning Galois modules. Let G = GK or GF be theabsolute Galois group, and let M be an arbitrary Z -module with a continuous
action of G. As usual, we write Z (1) = T (Gm ), and deﬁne the Tate twist M (r)of M for ∀r ∈ Z by :(r > 0)M ⊗ Z (r)M (r) = M(r = 0)Hom(Z (−r), M ) (r < 0)

as Galois modules. M → M (r) gives exact functors of Galois modules for ∀r ∈ Z,
and we have canonical isomorphisms M (r)(s) ∼= M (r + s) for ∀r, s ∈ Z. The
Galois action on Z (1) is via the cyclotomic character χ : G → Z×
 . The GaloisG
invariant and coinvariant is denoted respectively by M , MG .
Definition 3.3. The χ-part M χ of M is deﬁned by M χ = (M (−1)G )(1). It is
canonically identiﬁed with the maximal subgroup of M on which G acts via thecyclotomic character χ.
Note that the functors M → M G , M → M χ are both left exact.
3.2. Tate modules of the jacobians of curves. Now we return to the curve
X over a local ﬁeld K. Applying Lemma 2.5 and Lemma 3.1, we can restate theLemma 2.3 in the present case, in view of Remark 2.4, as follows :Lemma 3.4. For a proper smooth geometrically irreducible curve X over a local
field K which admits a K-rational point, there is a canonical short exact sequenceof abelian groups :/ J 0 [∞ ](−1)GF

0F/ J[∞ ](−1)GK/ (π ab (X)geo )∨
1

/0

ram 

∨
ab
geo ∨
where (π1ab (X)geo
ram ) denotes the -primary part of (π1 (X)ram ) .

Using the canonical isomorphism M (−1)G ∼= M χ (−1) of GF -modules, we can
express this lemma by the short exact sequence of GF -modules :
(3.1)

0

/ J 0 [∞ ]χ
F/ J[∞ ]χ

/ (π ab (X)geo )∨ (1)
1

ram 

/0

Now we relate this group with the component group Φ(J). For this we needthe description of Φ(J) by the Galois module associated to J, stated as follows :Lemma 3.5 ([SGA7] Exposé IX, Proposition 11.2). For an abelian variety A overa local field K, denote the Néron model and its special fiber respectively by A , AF .
Let A0F be the connected component of the identity, and denote the component
group by Φ(A) = AF /A0F . For any prime  = charF , there is a canonical isomorphism of GF -modules :Φ(A) ∼= Coker (T (A)I ⊗ (Q /Z ) −→ (T (A) ⊗ (Q /Z ))I )where Φ(A) is the -primary part of Φ(A), and (−)I denotes the invariant by
the action of the inertia group I ⊂ GK .

8

TERUYOSHI YOSHIDA

By the canonical isomorphism of GF -modules T (A)I ∼= T (A0F ) ([SGA7] Exposé
IX, Proposition 2.2.5, (2.2.3.3)), we can restate the above lemma by the short
exact sequence of GF -modules :(3.2)/ A0 [∞ ]0F/ A[∞ ]I/ Φ(A)

/0

Proof of Theorem 3.2. Applying the left exact functor M → M χ to the short
exact sequence (3.2) for A = J, we have the exact sequence :
(3.3)

0

/ J 0 [∞ ]χ
F/ J[∞ ]χ

/ Φ(J)χ


Comparing this with the short exact sequence (3.1), we have a canonical injection
χ
χ
∨
G
∼
(π1ab (X)geo
ram ) −→ Φ(J) (−1). As Φ(J) (−1) = Φ(J) (−1) injects to Φ(J) as
ﬁnite abelian groups and  = p was arbitrary, we have proven the theorem.
4. The p-primary part for low absolute ramification case
As in the previous section, we consider a proper smooth geometrically irreducible curve X over a local ﬁeld K which admits a K-rational point. Let e ∈ Ndenote the absolute ramification index of K, i.e. the normalized valuation of
p = charF in K. In this section, we prove the following :Theorem 4.1. Assume e < p − 1 and charK = 0. If X has semistable reduction
over OK , then (π1ab (X)geo
ram )p = 0, where (−)p denotes the p-primary part (p =charF ).
Remark 4.2. This theorem is a generalization of the Prop. 7 of Kato-Saito[6],where it is proved in the good reduction case. Here we employ a completely
diﬀerent method of proof.
1
Our task here is to analyze the Het(X, Z/pn Z)GK ∼= HomK−gp (µpn , Pic0 (X)[pn ])(Lemma 2.6). For this purpose, we recall the structure of the pn -torsion points
J[pn ] of the jacobian J = Pic0 (X), following [SGA7], Exposé IX. In the rest of
this section, we assume that X has semistable reduction over OK . In this case, J
has semistable reduction, i.e. the connected component of the identity JF0 of thespecial ﬁber JF of the Néron model J is a semiabelian scheme over F . Moreover,
if we denote by J 0 the connected component of the identity of J , we know thatJ 0 [pn ] is a quasi-ﬁnite group scheme over OK ([SGA7] Exposé IX, Lemme 2.2.1)

Now recall that any quasi-ﬁnite group  scheme Gf over OK has the canonical andf
G where G (the fixed part of G ) is ﬁnite
functorial decomposition G = G
ﬂat over OK and G  has empty special ﬁber ([EGA] II,(6.2.6), [SGA7] Exposé IX,(2.2.3.1)).
Coming back to our case, J 0 [pn ] has the decomposition :J 0 [pn ] = J 0 [pn ]f

J 0 [pn ]

CURVES OVER LOCAL FIELDS

9

where J 0 [pn ]f is a ﬁnite ﬂat group scheme. We denote the generic ﬁber of
J 0 [pn ]f by J 0 [pn ]f , which is a subgroup of J[pn ], the generic ﬁber of J 0 [pn ](recall (J 0 )K = J). Note that J 0 [pn ], J 0 [pn ]f have a common special ﬁber
JF0 [pn ], which is canonically isomorphic to Pic0 (XF )[pn ] by Lemma 3.1 (recall therunning hypothesis that X has a K-rational point).
Moreover, we have additional information by the semi-stability hypothesis([SGA7] Exposé IX, (5.5.8)) :J[pn ]/J 0 [pn ]f ∼= MK ⊗ (Z/pn Z) = (M ⊗ (Z/pn Z))K

(4.1)

where M denotes the character group of the toric part of JF , which is an unramiﬁed Galois module, free of ﬁnite rank over Z. We consider M as an étale groupscheme over OK , therefore justifying the notation MK . (Note that in general wehave to take the corresponding object for the dual abelian variety, but we havethe autoduality of the jacobian in the present case.)
Now we apply the result of Raynaud[12] to obtain the following lemma :Lemma 4.3. Under the hypothesis e < p − 1 and charK = 0, the following
natural homomorphism is an isomorphism :1
Het(XF , Z/pn Z)GF∼=

/ H 1 (X, Z/pn Z)GKet

Proof. By Lemma 2.6, we have :1(XF , Z/pn Z)GF ∼Het= HomF −gp (µpn , JF0 [pn ])H 1 (X, Z/pn Z)GK ∼= HomK−gp (µpn , J[pn ])etBy (4.1), we have an exact sequence :
0

/ Hom

K−gp (µpn , J

0[pn ]f )/ HomK−gp (µpn , J[pn ])/ HomK−gp (µpn , (M ⊗ (Z/pn Z))K )But by Corollaire 3.3.6 of Raynaud[12], we have :HomK−gp (µpn , (M ⊗ (Z/pn Z))K ) ∼= HomO −gp (µpn , M ⊗ (Z/pn Z)) = 0
K

which vanishes simply because µpn is connected and M ⊗(Z/pn Z) is étale. Therefore we have, using [12], Corollaire 3.3.6 again,
HomK−gp (µpn , J[pn ]) ∼= HomK−gp (µpn , J 0 [pn ]f ) ∼= HomOK −gp (µpn , J 0 [pn ]f )Therefore it suﬃces to see that HomOK −gp (µpn , J 0 [pn ]f ) ∼= HomF −gp (µpn , JF0 [pn ]),which is equivalent by Cartier duality ([SGA3] Exposé VIIA, (3.3.1)) to :HomOK −gp (J 0 [pn ]f ∗ , Z/pn Z) ∼= HomF −gp (JF0 [pn ]∗ , Z/pn Z)
where ∗ denotes the Cartier dual G ∗ = Hom(G , Gm ). But as Z/pn Z is étale, thisis clear by the Hensel’s lemma (e.g. [EGA] IV, (18.5.12)).

10TERUYOSHI YOSHIDA

Proof of Theorem 4.1. By Lemma 2.3, it is enough to show that the canonical11homomorphism Het(XF , Qp /Zp )GF −→ Het(X, Qp /Zp )GK is an isomorphism. But
by exactly the same argument as in (2.7) we have :


1
1
1
1
(XF , Qp /Zp )GF =Het(XF , Qp /Zp )[pn ]GF =Het(XF , n Z/Z)GF
Het
p
n
n


1
1
1
1
Het
(X, Qp /Zp )GK =Het(X, Qp /Zp )[pn ]GK =Het(X, n Z/Z)GKpnn
Therefore the desired isomorphism is deduced from the corresponding result in
the each p-power torsion level, namely the Lemma 4.3.
Combining with Th. 3.2, we have :Corollary 4.4. Assume e < p − 1 and charK = 0. If X admits a K-rational
point and has semistable reduction over OK , then there is an injection of finite∨
abelian groups (π1ab (X)geo
ram ) −→ Φ(J).
Remark 4.5. We have the canonical perfect dualiy :Φ ⊗ Φ −→ Q/Z
deduced by the autoduality of the jacobian and [SGA7] Exposé IX, (11.4.1) or
Conjecture 1.3 (proven in this case). This gives the canonical identiﬁcation Φ ∼=Φ∨ , which enables us to state Cor. 4.4 as the surjection Φ(J) −→ π1ab (X)geo.
ram
5. The case of the modular curve X0 (p)In this section, we apply the results of preceding section to the modular curve
X0 (p)/Qp for a prime p, which is a proper smooth geometrically irreducible curve
over Qp . For the deﬁnition and basic properties of the modular curve X0 (p), we
refer to Mazur [8]. The result of this section is summarized as follows :Theorem 5.1. π1ab (X0 (p)/Qp )geo has the following structure :0/ Φ(J0 (p))/ π ab (X (p)/Q )geo
1

0

p

/ r

Z

/0

geo
ab
where Φ(J0 (p)) ∼= π1ab (X0 (p)/Qp )geo
ram = π1 (X0 (p)/Qp )tor is a cyclic group of orderequal to the numerator of p−1, and r = g+h−1
where g is the genus of X0 (p) and122
h is the number of the supersingular points defined over Fp .

Remark 5.2. This result should be interpreted as the local analogue of the theorem of Mazur which asserts that π1ab (X0 (p)/Q)geo ∼= Φ(J0 (p)) ([8], Introduction,ab
geo
Theorem (2)). The fact that π1 (X0 (p)/Q) is equal to π1ab (X0 (p)/Qp )geo
ram =
(π1ab (X0 (p)/Qp )geo
shows
that
the
maximalabelianétalecoveringofX
0 (p)/Q
tor“uses up” all the ramiﬁcation allowed at the special ﬁber at p. The origin of thisphenomenon remains to be clariﬁed.

CURVES OVER LOCAL FIELDS

11

5.1. The part π1ab (X0 (p)/Qp )geo
ram . First, we apply the results of the preceding
sections to determine the part π1ab (X0 (p)/Qp )geo
ram . As e = 1 for Qp and X0 (p)admits a Qp -rational point (e.g. the ∞-cusp), and moreover X0 (p) has semistablereduction over Zp (Deligne-Rapoport[5], V-6, or Mazur[8], II-1), we know by Cor.
∨
4.4 that (π1ab (X0 (p)/Qp )geo
ram ) injects to Φ(J0 (p)), where J0 (p) is the jacobian ofX0 (p). By Mazur-Rapoport[9], we know that :Proposition 5.3 ([9], Theorem (A.1), b)). Φ(J0 (p)) is a cyclic group of order.
equal to the numerator of p−1
12Because the Shimura covering ([8], Cor. (2.3)) gives a cyclic étale covering of, which is by deﬁnition completely
X0 (p)/Qp of order equal to the numerator of p−1
12
ramiﬁed over one of the component of the special ﬁber, we have :Proposition 5.4. π1ab (X0 (p)/Qp )geo
ram is isomorphic to Φ(J0 (p)).
5.2. The part π1ab (X0 (p)/Fp )geo . We denote by X0 (p)/Zp the minimal regularmodel of X0 (p)/Qp following [8], and denote the special ﬁber by X0 (p)/Fp . Here
we determine the part π1ab (X0 (p)/Fp )geo , thereby completing the proof of Th. 5.1.
The result is :
 with the rank equal
Proposition 5.5. π1ab (X0 (p)/Fp )geo is a free module of over Z
g+h−1
to 2 , where g is the genus of X0 (p) and h is the number of the supersingularpoints defined over Fp .
Proof. Put G = Gal(Fp /Fp ). As each of the component of the special ﬁber
X0 (p)/Fp of X0 (p) is (geometrically) isomorphic to P1 , we know that Pic0 (X0 (p)/Fp )is a torus with the character group canonically isomorphic (as the G-module) tothe ﬁrst homology group H1 (Γ, Z) of the graph Γ of X0 (p)/Fp ([SGA7] Exposé
IX, 12.3). This graph is described in Mazur-Rapoport[9], §3. In particular, if
we denote the genus of X0 (p) by g, the total number of supersingular points are
g + 1, and H1 (Γ, Z) is a free module over Z of rank g.
Now by (2.5), (2.7), and Lemma 2.6, we have :∨π1ab (X0 (p)/Fp )geo ∼HomFp −gp (µN , Pic0 (X0 (p)/Fp )[N ])
=
N

∼
=



HomG (H1 (Γ, Z/N Z), Z/N Z)∨

N∼
Hom(H1 (Γ, Z/N Z)G , Q/Z)∨
= lim
←−
N

∼
H1 (Γ, Z/N Z)G
= lim
←−
N

∼

H1 (Γ, Z)G ⊗ Z/N Z ∼= lim= H1 (Γ, Z)G ⊗ Z
←−
N

12

TERUYOSHI YOSHIDA

where H1 (Γ, Z)G is the G-coinvariant of H1 (Γ, Z). The last isomorphism follows
from the fact that H1 (Γ, Z)G is clearly a ﬁnitely generated Z-module. Now it
only remains to determine the group H1 (Γ, Z)G explicitly.
As we know that the vertices of Γ is ﬁxed by G as the irreducible components
are deﬁned over Fp , and the edges of Γ which correspond to the supersingularpoints of X0 (p) are ﬁxed or interchanged by pairs by the Frobenius automorphismof G, according to whether the ﬁeld of deﬁnition of the corresponding supersingular point is Fp or Fp2 (these possibilities can be visibly read oﬀ from the Table
6 of [2]).
Therefore the Frobenius acts on the basis of the homology group formed by thepair of the edges corresponding to the pair of supersingular points deﬁned overFp2 by multipication by −1, which means that when we take the G-coinvariant,exactly these bases vanish. Therefore if we denote the number of the supersingularpoints deﬁned over Fp and the pairs of supersingular points deﬁned over Fp2
respectively by h and j (therefore g + 1 = h + 2j), the G-coinvariant of H1 (Γ, Z)
is a free module of rank g − j = g+h−1. (This rank is equal to the rank deﬁned
2
in the general setting in Saito[14], II-Def. 2.5.)

References[1] S. Abyhankar, Resolution of singularities for arithmetical surfaces, in : Arithmetical Algebraic Geometry (Harper and Row, 1963), pp.111-152.
[2] B. J. Birch, W. Kuyk, ed. Modular Functions of One Variable IV, Lecture Notes in Math.
476, Springer, 1975.
[3] S. Bloch, Algebraic K-theory and classfield theory for arithmetic surfaces, Annals of Math.
114 (1981), 229-265. (Erratum : Annals of Math. 126 (1987), 449.)[4] P. Deligne, Théorie de Hodge III, Publ. Math. I.H.E.S. 44 (1974), 5-77.
[5] P. Deligne, M. Rapoport, Schémas de modules des courbes elliptiques, in : Modular Functions of One Variable II (Springer LNM 349, 1973), pp.143-316.
[6] K. Kato, S. Saito, Unramified class field theory of arithmetical surfaces, Annals of Math.
118 (1983), 241-275.
[7] N. Katz and S. Lang, Finiteness theorems in geometric classfield theory, L’Enseignement
Math. 27 (1981), 285-319.
[8] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S. 47 (1977), 33-186.
[9] B. Mazur, M. Rapoport, Behavior of the Néron model of the jacobian of X0 (N ) at bad
primes, Appendix to [8].
[10] J. S. Milne, Étale Cohomology, Princeton Univ. Press, 1980.
[11] M. Raynaud, Spécialisation du foncteur de Picard, Publ. Math. I.H.E.S. 38 (1971), 27-76.
[12] M. Raynaud, Schéma en groupes de type (p, . . . , p), Bull. Soc. math. France 102 (1974),
241-280.
[13] J.-P. Serre, Groups algébriques et corps de classes, Hermann, 1959.
[14] S. Saito, Class Field Theory for Curves over Local Fields, J. of Number Theory 21 (1985),
44-80.
[15] T. Yoshida, Finiteness theorem in class field theory of varieties over local fields, preprint.

CURVES OVER LOCAL FIELDS

13[EGA] A. Grothendieck, J. Dieudonné, Éléments de Géométrie Algébrique, Publ. Math.
I.H.E.S. 4,8,11,17,20,24,28,32 (1960/61/63/64/65/66/67).
[SGA1] A. Grothendieck, et al., Revêtements Étales et Groupe Fondamental (SGA 1), Lecture
Notes in Math. 224, Springer, 1971.
[SGA3] A. Grothendieck, et al., Schémas en Groupes (SGA 3), Lecture Notes in Math.
151,152,153, Springer, 1970.
[SGA4] A. Grothendieck, et al., Théorie des Topos et Cohomologie Etale des Schémas (SGA
4), Lecture Notes in Math. 269,270,305, Springer, 1972/73.
[SGA7] A. Grothendieck, et al., Groupes de Monodromie en Géometrie Algébrique (SGA 7),
Lecture Notes in Math. 288,340, Springer, 1972/73.
Department of Mathematical Sciences, University of Tokyo
E-mail address: tyoshida@ms.u-tokyo.ac.jp

Preprint Series, Graduate School of Mathematical Sciences, The University of Tokyo
UTMS
2001–31 Tetsuhiro Moriyama: The mapping class group action on the homology of theconfiguration spaces of surfaces.
2001–32 Takeshi Katsura: On crossed products of the Cuntz algebra O∞ by quasi-freeactions of abelian groups.
2001–33 Yuichi Sugiki: The category of cosheaves and Laplace transforms.
2001–34 Hiroshige Kajiura: Homotopy algebra morphism and geometry of classical string
field theories.
2002–1 Tetsushi Ito: Stringy Hodge numbers and p-adic Hodge theory.
2002–2 Yasuyuki Kawahigashi and Roberto Longo: Classification of local conformal
nets. case c < 1.
2002–3 Takashi Taniguchi: A mean value theorem for orders of degree zero divisor classgroups of quadratic extensions over a function field.
2002–4 Shin-ichi Kobayashi: Iwasawa theory for elliptic curves at supersingular primes.
2002–5 Kazuya Kato and Fabien Trihan : Conjectures of Birch and Swinnerton-Dyer inpositive characteristics assuming the finiteness of the Tate-Shafarevich group.
2002–6 Satoshi Kondo: Euler systems on Drinfeld modular curves and zeta values.
2002–7 Teruyoshi Yoshida: Finiteness theorem in class field theory of varieties over
local fields.
2002–8 Teruyoshi Yoshida: Abelian étale coverings of curves over local fields and its
application to modular curves.

The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the former
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