2 MASANORI ASAKURA
supported on hyperplane section (cf. Thm. 3.3 (a) below). Next we showed that
the boundary map
:
CH2(X,
1; Ql/Zl) −→ Pic(Y ) Ql/Zl
is surjective (modulo finite groups) for X which has a good reduction Y . Thus
if Y contains primitive divisors, then
CH2(X,
1; Ql/Zl) contains indecomposable
elements and hence the map i cannot be surjective.
The technique used in the former step works also in case l = p. On the other
hand, in the latter step, we used the result of Sato-Saito [11], in which they proved
a weak Mordell-Weil type theorem for Chow group mod l different from p. Since
its mod p counterpart has not been obtained, we cannot use the same technique as
in [2] to show the surjectivity of the boundary map in case l = p.
Actually we do not need the surjectivity of to prove Theorem 1.1. It is
enough to show that the corank of the image of is greater than one. To do this,
we construct an indecomposable element in
CH2(X,
1;
Z/pnZ)
(never coming from
CH2(X,
1) Z/pnZ !). The strategy is as follows. We consider a quintic surface
X PQp
3
which contains an irreducible quintic curve C with four nodes. Let
˜
C C
be the normalization and {Pi,Qi} (1 i 4) the inverse images of the four nodes
on C. Since
˜
C is a curve of genus 2, there are (r1, ··· , r4) = (0, ··· , 0)
Zp4
and a rational function fn on
˜
C such that div(fn)
∑4
i=1
ri(Pi Qi) mod pn
(this is a simple application of the theorem of Mattuck [8] which asserts that the
Jacobian J(
˜)(Qp)
C is isomorphic to Zp
2
modulo finite groups). Thus the pair (C, fn)
determines an element in
CH2(X,
1; Z/pnZ). We then prove that its boundary is
nontrivial (hence indecomposable) under some assumptions.
This paper is organized as follows. In §2, we recall basic results on K-cohomology
and torsion of Chow groups of codimension 2. In §3, we prove a quintic surface
which satisfies all the conditions in §3.1 has infinite p-primary torsion in the Chow
group of 0-cycles. In §4, we prove the existence of such a quintic surface over Qp.
To do this we use Igusa’s j-invariants of hyperelliptic curves of genus two. We list
them in Appendix for the convenience of the reader.
The author would like to express sincere gratitude to Professors Shuji Saito
and Jean-Louis Colliot-Th´ el` ene for giving him many valuable comments.
2. Preliminaries
For an abelian group M we denote by M[n] (resp. M/n) the kernel (resp.
cokernel) of the multiplication by n. We denote the p-primary torsion by
M[p∞]
=
∪n≥1M[pn].
For schemes X and T over a base scheme S, we write X(T ) =
MorS(T, X) the set of S-morphisms, and say x X(T ) a T -valued point of X.
If T = SpecR, then we also write X(R) = X(SpecR) and say x X(R) a R-
rational point.
For a regular scheme X, let Zi(X) =
Zdim X−i(X)
be the free abelian group of
irreducible subvarieties of Krull dimension i.
2.1. K-cohomology and Gersten complex. Let X be a smooth variety
over a field F . Let us denote by
Xi
the set of irreducible subvarieties of X of
codimension i. We write the function field of Z
Xi
by ηZ .
Let Ki be the Zariski sheaf associated to a presheaf U Ki(U) where Ki(U) is
Quillen’s K-theory. The cohomology group
Hj(X,
Ki/n) is called the K-cohomology.
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