Contemporary Mathematics

Volume 571, 2012

http://dx.doi.org/10.1090/conm/571/11317

Quintic surface over p-adic local fields with infinite

p-primary torsion in the Chow group of 0-cycles

Masanori Asakura

Abstract. We construct a quintic surface over p-adic local fields such that

there is infinite p-primary torsion in the Chow group of 0-cycles.

1. Introduction

One of the long-standing problems on algebraic cycles is that for a projective

smooth variety X over a number field,

the Chow group

CHr(X)

of codimension r cycles is finitely generated Z-module?

In case r = 1, it has an aﬃrmative answer by the Mordell-Weil theorem together

with the fact that Neron-Severi groups are finitely generated Z-module. However

in case r ≥ 2, this is a widely open problem.

When the base field is a p-adic local field, the Chow group is no more finitely

generated Z-module in general. However, the torsion part is finite in case r = 1

([8]), and the same thing had been expected also in case r ≥ 2, until the first

counter-example was discovered by Rosenschon and Srinivas [10]. Soon after their

work, S. Saito and the author constructed another counter-example for 0-cycles on

a surface ([2]). We constructed a surface X over a p-adic local field such that the

l-primary torsion part CH0(X)[l∞] is infinite for l = p. On the other hand there

remained a question whether one can construct such an example for p-primary

torsion part of Chow group of 0-cycles. The purpose of this paper is to answer it:

Theorem 1.1. There is a quintic surface X ⊂ PQp

3

over Qp such that the p-

primary torsion part CH0(X ×Qp

K)[p∞]

is not finite for arbitrary finite extension

K of Qp.

Our proof is comparable with that of [2], however a new diﬃculty appears in

case l = p. Let us recall the outline of the proof of [2] briefly. It follows from the

universal coeﬃcient theorem on Bloch’s higher Chow group that we have the exact

sequence

(1.1) 0 −→

CH2(X,

1) ⊗ Ql/Zl

i

−→

CH2(X,

1; Ql/Zl) −→

CH2(X)[l∞]

−→ 0

for any l (possibly l = p). The proof of [2] breaks up into two steps. We first showed

that if X is generic then

CH2(X,

1) ⊗ Ql/Zl contains only decomposable elements

1991 Mathematics Subject Classification. Primary 14C25.

Key words and phrases. Algebraic cycles, higher Chow group, indecomposable elements.

c 2012 American Mathematical Society

1