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
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