Surfaces over a p-adic field with infinite torsion in the
Chow group of 0-cycles
Masanori Asakura and Shuji Saito

Contents
1 Introduction

12 Review of the cycle class map and `-adic regulator

4

3 Counterexample to the Bloch-Kato conjecture over p-adic field

7

4 Proof of Theorem 1.1

11

5 Appendix: SK1 of curves over p-adic fields

12

Abstract
We give an example of a projective smooth surface X over a p-adic field K suchthat for any prime ` different from p, the `-primary torsion subgroup of CH0 (X),the Chow group of 0-cycles on X, is infinite. A key step in the proof is disproving
a variant of the Bloch-Kato conjecture which characterizes the image of an `-adic
regulator map from a higher Chow group to a continuous étale cohomology of X byusing p-adic Hodge theory. With the aid of the theory of mixed Hodge modules, wereduce the problem to showing the exactness of the de Rham complex associatedto a variation of Hodge structure, which is proved by the infinitesimal method in
Hodge theory. Another key ingredient is the injectivity result on the cycle class
map for Chow group of 1-cycles on a proper smooth model of X over the ring ofintegers in K, due to K. Sato and the second author.

1

Introduction

Let X be a smooth projective variety over a base field K and let CHm (X) be the Chowgroup of algebraic cycles of codimension m on X modulo rational equivalence. In case K
is a number field, there is a folklore conjecture that CHm (X) is finitely generated, whichin particular implies that its torsion part CHm (X)tor is finite. The finiteness question
has been intensively studied by many authors, particularly for the case m = 2 and m =dim(X) (see nice surveys [27] and [8]).

1When K is a p-adic field (namely the completion of a number field at a finite place),
Rosenschon and Srinivas [29] have constructed the first example where CHm (X)tor isinfinite. They prove that there exists a smooth projective fourfold X over a p-adic fieldsuch that the `-torsion subgroup CH1 (X)[`] (see Notation) of CH1 (X), the Chow groupof 1-cycles on X, is infinite for each ` ∈ {5, 7, 11, 13, 17}.
A purpose of this paper is to give an example of a projective smooth surface X overa p-adic field such that for any prime ` different from p, the `-primary torsion subgroup
CH0 (X){`} (see Notation) of CH0 (X), the Chow group of 0-cycles on X, is infinite. Here
we note that for X as above, CH0 (X){`} is known to always be of finite cotype overZ` (namely the direct sum of a finite group and a finite number of copies of Q` /Z` .
The fact follows from Bloch’s exact sequence (2.3)). Thus our example presents infinite
phenomena of different nature from the example in [29]. Another noteworthy point isthat the phenomena discovered by our example happens rather generically.
To make it more precise, we prepare a notion of ‘generic surfaces’ in P3 . Let(d+3)(d+2)(d+1)/6−1
M ⊂ P(H 0 (P3Q , OP (d))) ∼= PQ

be the moduli space over Q of the nonsingular surfaces in P3Q (the subscription ‘Q’ indicatesthe base field), and let
f : X −→ M
be the universal family over M . For X ⊂ P3K , a nonsingular surface of degree d definedover a field K of characteristic zero, there is a morphism t : SpecK → M such thatX∼
= X ×M SpecK. We call X generic if t is dominant (i.e. t factors through the genericpoint of M ). In other words, X is generic if it is defined by an equation
X
aI z I (aI ∈ K)
F =I([z0 : z1 : z2 : z3 ] is the homogeneous coordinate of P3 , I = (i0 , · · · , i3 ) are multi-indicesand z I = z0i0 · · · z3i3 ) satisfying the following condition:(∗) aI 6= 0 for ∀I and {aI /aI0 }I6=I0 are algebraically independent over Q where I0 =(1, 0, 0, 0).
The main theorem is:
Theorem 1.1 Let K be a finite extension of Qp and X ⊂ P3K a nonsingular surface ofdegree d ≥ 5. Suppose that X is generic and has a projective smooth model XOK ⊂ P3OK
over the ring OK of integers in K. Let r be the Picard number (i.e. the rank of theNéron-Severi group) of the smooth special fiber of XOK . Then we haveCH0 (X){`} ∼= (Q` /Z` )⊕r−1 ⊕ (finite group)for ` 6= p.
One can construct a surface with infinite torsion in the Chow group of 0-cycles in thefollowing way. Let k be thePresidue field of K. Let Y be a smooth surface of degree d ≥ 5 in
P3k defined by an equation I cI z I (cI ∈ k) such that the Picard number r ≥ 2. Note that2

there exist such surfaces for each d. (For example if (p, d) = 1, one may choose a Fermat
type surface defined by z0d − z1d + z2d − z3d . Then the intersection of Y with the hyperplaneH ⊂ P3k defined by z0 − z1 is not irreducible so that r ≥ 2.) Take any liftings c̃I ∈ OK
and choose aI ∈ OK with ord(aI ) > 0 for each indices I such that {aI }I are algebraicallyindependent over Q(c̃I ), the subfield of KPgeneratedPover Q by c̃I for all I. Let X ⊂ P3K
be the surface defined by the equation I c̃I z I + I aI z I . Then it is clear that X isgeneric and has a smooth projective model over OK whose the special fiber is Y . Since
Y has the Picard number r ≥ 2, CH0 (X) has an infinite torsion subgroup by Theorem
1.1. It is proved in [28] that if the special fiber satisfies the Tate conjecture for divisors,
the geometric Picard number is congruent to d modulo 2. Thus if d is even, CH0 (X) has
an infinite torsion subgroup after a suitable unramified base change. Theorem 1.1 may
be compared with the finiteness results [9] and [28] on CH0 (X)tor for a surface X over ap-adic field under the assumption that H 2 (X, OX ) = 0 or, more generally, that the rank of
the Néron-Severi group does not change by reduction. For a nonsingular surface X ⊂ P3K
of degree d ≥ 1, the last condition is satisfied if d ≤ 3. Hence Theorem 1.1 leaves us aninteresting open question whether there is an example of a nonsingular surface X ⊂ P3K
of degree 4 for which CH0 (X){`} is infinite.
A distinguished role is played in the proof of Theorem 1.1 by the `-adic regulator map
1
ρX : CH2 (X, 1) ⊗ Q` −→ Hcont(Spec(K), H 2 (XK , Q` (2))) (XK = X ×K K)

from higher Chow group to continuous étale cohomology ([21]), where K is an algebraic
closure of K and ` is a prime different from ch(K). It is known that the image of ρX iscontained in the subspace
1(Spec(K), V ) (V = H 2 (XK , Q` (2)))
Hg1 (Spec(K), V ) ⊂ Hcont

introduced by Bloch and Kato [6]. In case ` 6= p this is obvious since Hg1 = H 1 bydefinition. For ` = p this is a consequence of a fundamental result in p-adic Hodge theory,
which affirms that every representation of GK = Gal(K/K) arising from the cohomology
of a variety over K is a de Rham representation (see the discussion after [6], (3.7.4)).
When K is a number field or a p-adic field, it is proved in [31] that CH2 (X){`} is finite
in case the image of ρX coincides with Hg1 (Spec(K), V ). Bloch and Kato conjecture thatit should be always the case if K is a number field.
The first key step in the proof of Theorem 1.1 is to disprove the variant of the BlochKato conjecture for a generic surface X ⊂ P3K over a p-adic field K (see Theorem 3.6). In
terms of Galois representations of GK = Gal(K/K), our result implies the existence of a
1-extension of Q` -vector spaces with continuous GK -action:(#)

0 → H 2 (XK , Q` (2)) → E → Ql → 0,such that E is a de Rham representation of GK but that there is no 1-extension of motives
over K:0 → h2 (X)(2) → M → h(Spec(K)) → 0
which gives rise to (#) under the realization functor. The rough idea of the proof of
the first key result is to relate the `-adic regulator map to an analytic regulator map
by using the comparison theorem for étale and analytic cohomology and then to show3

that the analytic regulator map is the zero map. With the aid of the theory of mixed
Hodge modules ([33]), this is reduced to showing the exactness of the de Rham complex
associated to a variation of Hodge structure, which is proved by the infinitesimal methodin Hodge theory. This is done in §3 after in §2 we review some basic facts on the cycle
class map for higher Chow groups.
Another key ingredient is the injectivity result on the cycle class map for the Chowgroup of 1-cycles on a proper smooth model of X over the ring OK of integers in K due toSato and the second author [30]. It plays an essential role in deducing the main theorem
1.1 from the first key result, which is done in §4.
Finally, in §5 Appendix, we will apply our method to produce an example of a curve
C over a p-adic field such that SK1 (C)tor is infinite.
Acknowledgment. The authors are grateful to Dr. Kanetomo Sato for stimulating discussions and helpful comments. The authors thank the referee for constructive comments
which have improved the presentation of our paper very much.
Notations. For an abelian group M , we denote by M [n] (resp. M/n) the kernel (resp.
cokernel) of multiplication by n. For a prime number ` we put[
MM {`}.
M {`} :=
M [`n ], Mtor :=n`

For a nonsingular variety X over a field CHj (X, i) denotes Bloch’s higher Chow groups.
We write CHj (X) := CHj (X, 0) for the (usual) Chow groups.

2

Review of the cycle class map and `-adic regulator

In this section X denotes a smooth variety over a field K and n denotes a positive integer
prime to ch(K).
2.1: By [18] we have the cycle class map
2i−j
i
ci,j
ét : CH (X, j, Z/nZ) → Hét (X, Z/nZ(i)),

where the right hand side is the étale cohomology of X with coefficients µ⊗i
n , Tate twistof the sheaf of n-th roots of unity. The left hand side is Bloch’s higher Chow group withfinite coefficient which fits into the exact sequence
0 → CHi (X, j)/n → CHi (X, j, Z/nZ) → CHi (X, j − 1)[n] → 0.

(2.1)

In this paper we are only concerned with the map
2,1
3
cét = cét
: CH 2 (X, 1, Z/nZ) → Hét
(X, Z/nZ(2)).

(2.2)

By [7] it is injective and its image is equal to33
3
N Hét(X, Z/nZ(2)) = Ker Hét
(X, Z/nZ(2)) → Hét(Spec(K(X)), Z/nZ(2)),

where K(X) is the function field of X. In view of (2.1) it implies an exact sequencecét
30 −→ CH2 (X, 1)/n −→N Hét(X, Z/nZ(2)) −→ CH2 (X)[n] −→ 0.

4

(2.3)

2.2: We also need the cycle map to continuous étale cohomology group (cf. [21]):3
ccont : CH2 (X, 1) −→ Hcont(X, Z` (2)),

where ` is a prime different from ch(K). Note that in case K is a p-adic field we have
3
3
(X, Z` (2)) = lim Hét
(X, Z/`n Z(2))
Hcont
←−
n

and ccont is induced by cét by passing to the limit. We have the Hochschild-Serre spectral
sequencei+ji
E2i,j = Hcont(Spec(K), H j (XK , Z` (2))) ⇒ Hcont(X, Z` (2)).
(2.4)
If K is finitely generated over the prime subfield and X is proper smooth over K, the Weilconjecture proved by Deligne implies that H 0 (Spec(K), H 3 (XK , Q` (2))) = 0. The sameconclusion holds if K is a p-adic field and X is proper smooth having good reduction
over K (In case ` 6= p this follows from the proper smooth base change theorem forétale cohomology. In case ` = p one uses comparison theorems between p-adic étale andcrystalline cohomology and the Weil conjecture for crystalline cohomology). Thus we getunder these assumptions the following map
1
ρX : CH2 (X, 1) −→ Hcont(Spec(K), H 2 (XK , Q` (2)))(2.5)

as the composite of ccont ⊗ Q` and an edge homomorphism
3
1
Hcont
(X, Q` (2)) → Hcont(Spec(K), H 2 (XK , Q` (2))).

2.3: For later use, we need an alternative definition of cycle class maps. For an integeri ≥ 1 we denote by Ki the sheaf on XZar , the Zariski site on X, associated to the presheaf
U 7−→ Ki (U ). By [23], 2.5, we have canonical isomorphisms
1
CH2 (X, 1) ' HZar
(X, K2 ),

1CH2 (X, 1, Z/nZ) ' HZar
(X, K2 /n).

(2.6)

Let ét : Xét → XZar be the natural map of sites and putiHét(Z/nZ(r)) = Ri ∗ét µ⊗rn .

The universal Chern classes in the cohomology groups of the simplicial classifying spacefor GLn (n ≥ 1) give rise to higher Chern class maps on algebraic K-theory (cf. [17],[35]). It gives rise to a map of sheaves:i
Ki /n −→ Hét
(Z/nZ(i)).

(2.7)

By [26] it is an isomorphism for i = 2 and induces an isomorphism
∼
=

1
1
2
HZar
(X, K2 /n) −→ HZar
(X, Hét(Z/nZ(2))).

(2.8)

By the spectral sequencepq
p+q
E2pq = HZar
(X, Hét(Z/nZ(2))) =⇒ Hét
(X, Z/nZ(2)).
qptogether with the fact HZar
(X, Hét(Z/nZ(2))) = 0 for p > q shown by Bloch-Ogus [7], we
get an injective map
1
2
3(X, Z/nZ(2))
HZar(X, Hét(Z/nZ(2))) −→ Hét

Again by the Bloch-Ogus theory the image of the above map coincides with the coniveau
3
filtration N Hét
(X, Z/nZ(2)). Combined with (2.6) and (2.8) we thus get the map
∼
=

∼
=

⊂

1
3
3
(X, Z/nZ(2)) −→ Hét
(X, Z/nZ(2)).
cét : CH2 (X, 1, Z/nZ) −→ HZar
(X, K2 /n) −→ N Hét

It is known that this agrees with the map (2.2) (see [10] Proposition 1).
5

2.4: Now we work over the base field K = C. Let Xan be the site on the underlying
analytic space X(C) endowed with the ordinary topology. Let an : Xan → XZar be thenatural map of sites and put
√
i
Z(r)
(Z(r)
=
(2π
(Z(r)) = Ri anHan−1)r Z).
∗
Higher Chern class map then gives a map of sheavesiKi −→ Han
(Z(i)).

(2.9)

By the same argument as before, it induces a map∼=

3
1(X(C), Z(2))(X, K2 ) −→ Han
can : CH2 (X, 1) −→ HZar

(2.10)

3(X(C), C), the Hodge filtration deLemma 2.1 The image of can is contained in F 2 Han
fined in [14]. In particular if X is complete, the image is torsion.
r
Proof. Let HD(Z(i)) be the sheaf on XZar associated to a presheafr
U 7→ HD(U, Z(i))•
where HDdenotes Deligne-Beilinson cohomology (cf. [15], 2.9). Higher Chern class maps2
to Deligne-Beilinson cohomology give rise to the map K2 → HD
(Z(2)) and can factors as
in the following commutative diagram
1
1
2
1
2
HZar
(X, K2 ) −−−→ HZar
(X, HD(Z(2))) −−−→ HZar
(X, Han(Z(2)))




ay
y
3
(X, Z(2))HDb

3(X(C), Z(2)).
−−−→ Han

Here the map a is induced from the spectral sequencepq
p+q
E2pq = HZar
(X, HD(Z(2))) =⇒ HD(X, Z(2))p11(Z(2)) ∼in view of the fact HZar
(Z(2))) = 0 for ∀p > 0 since HD(X, HD= C/Z(2) (constant
2 3
sheaf). Since the image of b is contained in F Han (X(C), C) (see [15], 2.10), so is theimage of can . Q.E.D.

Lemma 2.2 We have the following commutative diagram
CH2 (X, 1)


y

c

an
3
(X(C), Z(2))
−−−
→ Han


y

(2.11)cét
CH2 (X, 1, Z/nZ) −−−→ Hét3 (X, Z/nZ(2))

Here the right vertical map is the composite
∼
=

3
3
Han
(X(C), Z(2)) → Han(X(C), Z(2) ⊗ Z/nZ) −→ Hét3 (X, Z/nZ(2))

and the isomorphism comes from the comparison isomorphism between étale cohomology
and ordinary cohomology (SGA4 21 , Arcata, 3.5) together with the isomorphism
Z(1) ⊗ Z/nZ ' (an )∗ µn
given by the exponential map.
6

Proof. This follows from the compatibility of (2.7) and (2.9), namely the commutativityof the diagrami(Z(i))
Ki −−−→ Han




y
y
i
Ki /n −−−→ Hét
(Z/nZ(i)),

and it follows from the compatibility of the universal Chern classes ([17] and [35]). Q.E.D.

3Counterexample to the Bloch-Kato conjecture overp-adic field

In this section K denotes a p-adic field and let X be a proper smooth surface over K. We
fix a prime ` (possibly ` = p) and consider the map (2.5)
12
ρX : CH2 (X, 1) −→ Hcont(Spec(K), V ) (V = Hét
(XK , Q` (2)))).

(3.1)

Define the primitive part Ve of V by:
2
Ve := Hét
(XK , Q` (2))/V0 ,V0 = [HX ] ⊗ Q` (1),

(3.2)

2
where [HX ] ∈ Hcont(XK , Q` (1)) is the cohomology class of a hyperplane section. Noting∪[HX ]
4
2(XK , Q` (3)) ,Ve ' Ker Hét
(XK , Q` (2)) −→ Hét
we get a decomposition as GK -modules:
V = Ve ⊕ V0 .

(3.3)

Let
1
ρe : CH2 (X, 1) −→ Hcont(Spec(K), Ve )be the induced map.
Theorem 3.1 Let X ⊂ P3K be a generic smooth surface of degree d ≥ 5. Then ρe is thezero map for arbitrary `.
Remark 3.2 (1) This is an analogue of [38] 1.6 (where she worked on Deligne-Beilinson
cohomology).
(2) Bloch-Kato [6] considers regulator maps such as (3.1) for a smooth projective variety
over a number field and conjectures that its image coincides with Hg1 . We will see
later (see Theorem 3.6) that the variant of the conjecture over a p-adic field is false
in general.
(3) The construction of a counterexample mentioned in (2) hinges on the assumptionthat the surface X ⊂ P3K is generic. One may still ask whether the image of l-adic
regulator map coincides Hg1 for a proper smooth variety X over a p-adic field when
X is defined over a number field.
7

Proof. Let f : X → M be as in the introduction and let t : Spec(K) → M be a dominantmorphism such that X ' X ×M Spec(K). For a morphism S → M of smooth schemes
over Q let fS : XS = X ×M S → S be the base change of f . The same construction of(2.5) give rise to the regulator map
1ρS : CH2 (XS , 1) → Hcont(S, VS ),where VS = R2 (fS )∗ Ql (2) is a smooth Ql -sheaf on S. Define the primitive part of VS :VeS = R2 (fS )∗ Ql (2)/[H] ⊗ Ql (1),where [H] ∈ H 0 (S, R2 (fS )∗ Ql (1)) is the class of a hyperplane section. Let
1
ρeS : CH2 (XS , 1) → Hcont(S, VeS ),be the induced map. Note
CH2 (X, 1) = lim CH2 (XS , 1),−→S

where S → M ranges over the smooth morphisms which factor t : Spec(K) → M . Notealso that we have the commutative diagram for such S:
ρeS
CH2 (XS , 1) −−−
→


y

1
(S, VeS )
Hcont


y

ρe
1
(Spec(K), Ve ).
CH2 (X, 1) −−−→ Hcont

Thus it suffices to show
1
(S, VeS ) = 0.
Hcont

Without loss of generality we suppose S is an affine smooth variety over a finite extension
L of Q.
Claim 3.3 Assume d ≥ 4. The natural map
1
Hcont(S, VeS ) −→ Hét1 (SQ , VeS )(SQ := S ×L Spec(Q))

is injective.
0
Indeed, by the Hochschild-Serre spectral sequence, it is enough to see Hét
(SQ , VeS ) = 0,
which follows from [3], Th.6.1. (2).

By SGA4 12 , Arcata, Cor.(3.3) and (3.5.1) we have
11
1
(SC , VeS ) ' Han(S(C), VeSan ) ⊗ Ql ,Hét
(SQ , VeS ) ∼= Hét(SC := S ×L Spec(C))

where VeSan is the primitive part of VSan = R2 (fSan )∗ Q(2) with fSan : (XSC )an → (SC )an ,the natural map of sites. By definition VeSan is a local system on S(C) whose fiber over2s ∈ S(C) is the primitive part of Han(Xs (C), Q(2)) for Xs , the fiber of XS → S over s.
Due to Lemma 2.2, it suffices to show the triviality of the image of the map
2
1
e an
ρean
S : CH (XSC , 1) −→ Han (S(C), VS )8

which is induced from3
(XS (C), Q(2))
can : CH2 (XSC , 1) −→ Han

by using the natural map
31
Han(XS (C), Q(2)) → Han(S(C), VSan )arising from the Leray spectral sequence for fSan : (XSC )an → (SC )an and the vanishing
R3 (fSan )∗ Q(2) = 0.
2 1e an
Claim 3.4 The image of ρean
S is contained in the Hodge filtration F Han (S(C), VS ⊗ C)
defined by the theory of Hodge modules [33] §4.

This follows from the functoriality of Hodge filtrations and Lemma 2.1.
1
It is quite complicated to describe the Hodge filtration on Han(S(C), VeSan ⊗C) precisely.
However, all that we need is the following property:
Claim 3.5 For integers m, p ≥ 0 there is a natural injective mapmm
F p Han(S(C), VeSan ⊗ C) → HZar
(SC , Gp DR(VeSan ))where Gp DR(VeSan ) is the complex of Zariski sheaves on SC :2
2
F p HdR(XS /S)prim ⊗ OSC → F p−1 HdR(XS /S)prim ⊗ Ω1SC /C → · · ·22· · · → F p−r HdR(XS /S)prim ⊗ ΩrSC /C → F p−r HdR(XS /S)prim ⊗ Ωr+1
SC /C → · · ·••Here HdR(XS /S) denotes the de Rham cohomology of XS /S, and HdR(XS /S)prim is its
primitive part defined by the same way as before, and the maps are induced from theGauss-Manin connection thanks to Griffiths transversality.

This follows from [1] Lemma 4.2. We note that its proof hinges on the theory of mixed
Hodge modules. Key points are Deligne’s comparison theorem for algebraic and analytic
cohomology of a vector bundle with integrable connection with regular singularities (cf.
[12], §6) and degeneration of the Hodge spectral sequence for cohomology with coefficients(cf. [33], (4.1.3)).
By the above claims we are reduced to showing the exactness at the middle term of
the following complex:
22
2
F 2 HdR(XS /S)prim ⊗ OSC → F 1 HdR(XS /S)prim ⊗ Ω1SC /C → HdR(XS /S)prim ⊗ Ω2SC /C . (3.4)

This is proved by the infinitesimal method in Hodge theory. We sketch the proof. Let
f : XS → S be the natural morphism. The assertion follows from the exactness at the
middle term of the complex:f∗ Ω2XS /S ⊗ OSC −→ (R1 f∗ Ω1XS /S )prim ⊗ Ω1SC /C −→ R2 f∗ OXS ⊗ Ω2SC /C
9

(3.5)

and the injectivity of the complex:f∗ Ω2XS /S ⊗ Ω1SC /C −→ (R1 f∗ Ω1XS /S )prim ⊗ Ω2SC /C .

(3.6)

These complexes are induced by the complex (3.4) by Griffiths transversality. In case S =M ⊂ P(H 0 (P3Q , OP (d))), these assertions are proved as follows: Let P = C[z0 , z1 , z2 , z3 ]and P n ⊂ P be the subspace of the homogeneous polynomials of degree n. Take apoint x ∈ M (C) and choose F ∈ P d which defines the surface corresponding to x. Let∂F∂F
R = C[z0 , z1 , z2 , z3 ]/( ∂z, · · · , ∂z
) be the Jacobian ring and Rn ⊂ R be the image of P n
03
in R. Then the fibers over x of (3.5) and (3.6) are identified with the following Koszul
complexes:2

Rd−4 −→ R2d−4 ⊗ (Rd )∗ −→ R3d−4 ⊗ ∧(Rd )∗ ,2

Rd−4 ⊗ (Rd )∗ −→ R2d−4 ⊗ ∧(Rd )∗

(3.7)
(3.8)

where (Rd )∗ denotes the dual space of R and the maps are induced from the multiplication
R ⊗ R → R. Then the Donagi symmetrizer lemma (cf. [19], p.76) implies that (3.7) isexact at the middle term if d ≥ 5 and (3.8) is injective if d ≥ 3, which proves the desired
assertion in case S = M . The assertion in case S is dominant over M is reduced to thecase S = M by an easy argument (cf. [4] §9). This completes the proof of Theorem 3.1.
Q.E.D.
Let OK ⊂ K be the ring of integers and k be the residue field. In order to construct
an example where the image of the regulator map
ρX
12
ρX : CH2 (X, 1) −→Hcont(Spec(K), V ) (V = Hét
(XK , Q` (2))))

is not equal to Hg1 (Spec(K), V ), we now take a proper smooth surface X having good
reduction over K so that X has a proper smooth model XOK over Spec(OK ). We denotethe special fiber by Y . By [24] (see the diagram below 5.7 on p.341), there is a commutative
diagramρe
CH2 (X, 1) −−−→
Hg1 (Spec(K), V )




(3.9)
∂
y
y
α

1
(Spec(K), V )/Hf1 (Spec(K), V )CH1 (Y ) −−−→ Hcont
1
where Hf1 ⊂ Hg1 ⊂ Hcont
are the subspaces introduced by Bloch-Kato [6] and ∂ is aboundary map in localization sequence for higher Chow groups.

Theorem 3.6 Let X ⊂ P3K be a generic smooth surface of degree d ≥ 5. Assume that X
has a projective smooth model XOK ⊂ P3OK over OK and let Y ⊂ P3k be its special fiber.
(1) The image of ∂ ⊗ Q is contained in the subspace of CH1 (Y ) ⊗ Q generated by theclass [HY ] of a hyperplane section of Y .
(2) Let r be the Picard number of Y . Then
dimQ` Hg1 (Spec(K), V )/Image(ρX ) ≥ r − 1.
10

Proof. Recall V = Ve ⊕ V0 , a decomposition as GK -modules (cf. (3.3)). Let W ⊂CH 2 (X, 1) be the image of Z · [HX ] ⊗ K × under the product map CH1 (X) ⊗ K × →CH2 (X, 1). Then it is easy to see ρX induces an isomorphism
1
W ⊗ Q` ' Hg1 (Spec(K), V0 ) = Hcont(Spec(K), V0 )and that ∂(W ) = Z · [HY ] ⊂ CH1 (Y ). Hence (1) follows from Theorem 3.1 together with
injectivity of α in (3.9) proved by [24], Lemma 5-7.
1(Spec(K), V0 )/Hf1 (Spec(K), V0 ) = 1 (see [6],As for (2) we first note that dimQ` Hcont
3.9). Moreover the same argument (except using the Tate conjecture) in the last part of§5 of [24] shows
dimQ` (CH1 (Y ) ⊗ Q` ) ≤ dimQ` Hg1 (Spec(K), V )/Hf1 (Spec(K), V ) .
Hence (2) follows from (1). Q.E.D.

Remark 3.7 Let the assumption be as in Corollary 3.6. Then we have(
r−1
dimQ` Hg1 (Spec(K), V )/Image(ρX ) ≥r − 1 + (h0,2 + h1,1 − 1)[K : Qp ]` 6= p`=pwhere hp,q := dimK H q (X, ΩpX/K ) denotes the Hodge number. Moreover the equality holds
if and only if the Tate conjecture for divisors on Y holds. This follows from Theorem 3.1
and the computation of dimQ` Hg1 (Spec(K), V ) using [6] 3.8 and 3.8.4. The details areomitted.

4

Proof of Theorem 1.1

Let K be a p-adic field and OK ⊂ K the ring of integers and k the residue field. Let us
consider schemes
j
X
−−−→
XOK
←−−−
Y
i






(4.1)
y
y
y
Spec(K) −−−→ Spec(OK ) ←−−− Spec(k)where all vertical arrows are projective and smooth of relative dimension 2 and the diagrams are Cartesian. We have a boundary map in localization sequence for higher Chowgroups with finite coefficients∂ : CH2 (X, 1, Z/nZ) → CH1 (Y )/n.
For a prime number `, it induces∂` : CH2 (X, 1, Q` /Z` ) → CH1 (Y ) ⊗ Q` /Z` ,where CH2 (X, 1, Q` /Z` ) := lim CH2 (X, 1, Z/`n Z).
−→
n

11

Theorem 4.1 For ` 6= p := ch(k), ∂` is surjective and has finite kernel. Hence we have
CH2 (X, 1, Q` /Z` ) ∼= (Q` /Z` )⊕r + (finite group)
where r is the rank of CH1 (Y ).
Theorem 1.1 is an immediate consequence of Theorems 3.6 (1), 4.1, and the exact
sequence (2.1):
0 → CH2 (X, 1) ⊗ Q` /Z` → CH2 (X, 1, Q` /Z` ) → CH2 (X){`} → 0.
Proof. Write Λ = Q` /Z` We have a commutative diagram:
∂

i

j∗

∂

i

j∗

∗
→ CH2 (XOK ) ⊗ Λ −−−→ CH2 (X) ⊗ Λ
CH2 (X, 1, Λ) −−−→ CH1 (Y ) ⊗ Λ −−−




c1
c2
c3
c4
y
y
y
y
∗
ét
ét
4
4
2
3
(XOK , Λ(2)) −−−(X, Λ(2))→ Hét→ Hét(Y, Λ(1)) −−−(X, Λ(2)) −−−→ HétHétHere the upper exact sequence arises from localization theory for higher Chow groups
with finite coefficient ([25], Theorem 1.7) and the lower from localization theory for étale
cohomology together with absolute purity [16]. The vertical maps are étale cycle class
maps. By (2.3) c1 is injective. Noting CH1 (Y ) = H 1 (Y, Gm ), c2 is injective by theKummer theory. It is shown in [30] that c3 is an isomorphism. Hence the diagram∗
reduces the proof of Theorem 4.1 to showing that Ker(∂ét ) and Ker(jét) are finite. Thisis an easy consequence of the proper base change theorem for étale cohomology and theWeil conjecture ([13]). For the former we use also an exact sequence
∂

ét
3
3
2
Hét
(XOK , Λ(2)) → Hét
(X, Λ(2)) −→Hét(Y, Λ(1)).

Q.E.D.

5

Appendix: SK1 of curves over p-adic fields

Let C be a proper smooth curve over a field K and consider CH2 (C, 1). By [23], 2.5, wehave an isomorphism
1CH2 (C, 1) ' HZar
(C, K2 ) ' SK1 (C).
By definitionδ

SK1 (C) = Coker(K2 (K(C)) −→M

K(x)× ),

x∈C0

where K(C) is the function field of C, C0 is the set of the closed points of C, and K(x)is the residue field of x ∈ C0 , and δ is given by the tame symbols. The norm maps
K(x)× → K × for x ∈ C0 induce
NC/K : SK1 (C) → K × .
We write V (C) = Ker(NC/K ).
12

When K is a p-adic field, it is known by class field theory for curves over local field([34]), that V (C) is a direct sum of its maximal divisible subgroup and a finite group. Aninteresting question is whether the divisible subgroup is uniquely divisible, or equivalently
whether SK1 (C)tor is finite. In case the genus g(C) = 1 affirmative results have beenobtained in [32] and [2]. The purpose of this section is to show that the method in theprevious sections gives rise to an example of a curve C of g(C) ≥ 2 such that SK1 (C)tor
is infinite.
Let C be as in the beginning of this section and let n be a positive integer prime toch(K). We have the cycle class map
2
cét : CH2 (C, 2, Z/nZ) → Hét
(C, Z/nZ(2)).

(5.1)

The main result of [26] implies that the above map is an isomorphism. In view of theexact sequence (cf. (2.1)):
0 → CH2 (C, 2)/n → CH2 (C, 2, Z/nZ) → SK1 (C)[n] → 0,
we get the exact sequence ([37] 23.4):
2
0 → CH2 (C, 2)/n → Hét
(C, Z/nZ(2)) → SK1 (C)[n] → 0.

(5.2)

We will use also cycle class map to continuous étale cohomology:
2
ccont : CH2 (C, 2) ⊗ Q` → Hcont(C, Q` (2))

where ` is any prime number different from ch(K). When K is a p-adic field, one easily
shows
2
1
1
(5.3)
Hcont
(C, Q` (2)) ' Hcont(Spec(K), Hét
(CK , Q` (2)))
by using the Hochschild-Serre spectral sequence (2.4). Hence we get the map
11
ρC : CH2 (C, 2) ⊗ Q` → Hcont(Spec(K), Hét
(CK , Q` (2))).

(5.4)

Note that ρC is trivial if C has good reduction and ` 6= p, since the group on the righthand side is trivial. The last fact is a consequence of the proper smooth base change
theorem for étale cohomology and the weight argument.
Let Mg be the moduli space of tri-canonically embedded projective nonsingular curvesof genus g ≥ 2 over the base field Q (cf. [11]), and let f : C → Mg be the universal family.
Definition 5.1 Let C be a proper smooth curve over a field K of characteristic zero.
We say C is generic if there is a dominant morphism Spec(K) → Mg such that C ∼=C ×Mg Spec(K).
Theorem 5.2 Let K be a p-adic field and let C be a generic curve of genus g ≥ 2 over
K. Then ρC is the zero map for all `. We have an isomorphismSK1 (C)tor ∼H 2 (C, Z/nZ(2)) .
= Hét2 (C, Q/Z(2)) := lim−→ étnRemark 5.3 Theorem 5.2 is comparable with the main result of [20] where they worked
on Deligne-Beilinson cohomology.
13

Proof. The second assertion follows easily from the first in view of (5.2). The first assertionis shown by the same method as the proof of Theorem 3.1, noting the following fact (cf.
[20] §3): Let S → Mg be a dominant smooth morphism and put f : CS := C ×Mg S → S.
Then the mapf∗ Ω1CS /S −→ R1 f∗ OCS ⊗ Ω1S/Q
induced from the Gauss-Manin connection is injective. Q.E.D.
Corollary 5.4 Let C be as in Theorem 5.2. Assume the Jacobian variety J(C) has
semistable reduction over K. Let J be the Neron model of J with Js , its special fiber. Letr be the dimension of the maximal split torus in Js . For a prime `, we haveSK1 (C){`} ' (Q` /Z` )r` ⊕ (finite group),
where r` = r for ` 6= p and rp = r + 2g[K : Qp ].
For example SK1 (C){`} is infinite for any ` if C is a Mumford curve (i.e. a propersmooth curve with semistable reduction over K such that the irreducible components are
isomorphic to P1k and that they intersect each other at k-rational points, where k is theresidue field of K), which is generic in the sense of Definition 5.1.
Theorem 5.4 follows from Theorem 5.2 and the following:
Lemma 5.5 Let C be proper smooth curve over a p-adic field K Assume J(C) has
semistable reduction over K and let r` be as above. Then21
dimQ` Hcont(C, Q` (2)) = dimQ` Hcont(Spec(K), V ) = r` . V = Hét1 (CK , Q` (2)) .
Proof. The first equality follows from (5.3). By [22], Th. 5 and Cor. 7 (p. 354–355) we0
2have Hcont(Spec(K), V ) = 0 and dimQ` Hcont(Spec(K), V ) = r. Lemma 5.5 now follows
from computation of Euler-Poincaré characteristic given in [36], II 5.7. Q.E.D.
Remark 5.6 Using [6], 3.8.4 together with a well-known fact on the structure as Gal(K/K)module of the Tate module of an abelian variety over K (cf. [39], Exposé IX), one can show
1
(Spec(K), V ) = Hg1 (Spec(K), V ). Hence, if C is a generic curve of genus≥ 2,that Hcont
then the map ρC (5.4) does not surject on to Hg1 if r` ≥ 1. This gives another counterexample to a variant of the Bloch-Kato conjecture for p-adic fields.

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Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, JAPANE-mail address : asakura@math.kyushu-u.ac.jp
Graduate School of Mathematical Sciences, Tokyo University, Tokyo 153-8914, JAPANE-mail address : sshuji@msb.biglobe.ne.jp17