CONJECTURE OF COLLIOT-THÉLÈNE ON ZERO-CYCLES OVER
LOCAL FIELDS: ABSTRACTSHUJI SAITOLet X be a smooth projective variety over a field k. Let CH0 (X) be the Chow groupof the zero-cycles on X modulo rational equivalence and let A0 (X) ⊂ CH0 (X) be thesubgroup of cycle classes of degree 0. Let
albX : A0 (X) → AlbX (k)be the map induced by the Albanese map X → AlbX , where AlbX (k) is the group of thek-rational points of AlbX .
By Abel’s theorem albX is injective if dim(X) = 1. In higher dimension, Mumford discovered that the situation is rather chaotic: Ker(albX ) may be too large to be understoodby a standard geometric structure.
On the other hand, the situation in case k is a field of arithmetic nature presents astriking contrast to the above case. If k is finite, Ker(albX ) has been explicitly determinedby geometric class field theory. One of the most fascinating and challenging conjecturesin arithmetic geometry is a conjecture of Bloch and Beilinson that Ker(albX ) is finite incase k is a number field.
In this talk we focus on the case where k is a p-adic field. In this case Colliot-Thélèneposed the following questions ([1]):(CT 1)

Ker(albX ) = D(X)M(finite) ??

Here D(X) is the maximal divisible subgroup of A0 (X). By a known result on the structureof AlbX (k), (CT1) implies(CT 1)0A0 (X) = D(X)M(finite) ??

where D0 (X) is the maximal subgroup of A0 (X) divisible by all integers prime to p.
Another question of Colliot-Thélène is the following.
(CT 2)

D(X)tor = 0 ??

Note(CT 1) + (CT 2) ⇒ CH0 (X)tor is finite
There have been several affirmative results on the questions for special varieties; surfaces,varieties fibered over curves in quadrics or Severi-Brauer varieties, products of curves,
rationally connected varieties (see [1]. Some of them are mentioned in the talk).
In this talk I present an affirmative result on (CT1) and a negative result on (CT2).
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SHUJI SAITOThe affirmative result (joint work with K. Sato, [3]) affirms that (CT1)0 is true if X hasa regular projective flat model X over the ring Ok of integers in k such that the reduced
part of its special fiber is a simple normal crossing divisor on X .
The negative result (joint work with M. Asakura, [2]) provides an example of a smoothsurface X ⊂ P3k such that CH0 (X){`}, the `-primary torsion part of CH0 (X), is infinite
for all prime ` 6= p.
The affirmative result is deduced from a theorem which claims that the étale cycle class
map for Chow group of 1-cycles on the model X of X over Ok is an isomorphism.
Main ingredients in its proof are:the Bertini theorem for schemes over a discrete valuation ring by Jannsen-Saito,
the absolute purity theorem of Gabber,
the Weil conjecture proved by Deligne,
an affine vanishing theorem of Artin-Gabber and its refinement.
A key step in the proof of the negative result is disproving a variant of the Bloch-Katoconjecture which characterizes the image of an `-adic regulator map from a higher Chowgroup to a continuous étale cohomology of X by using p-adic Hodge theory. With the aidof the theory of mixed Hodge modules, we reduce the problem to showing the exactnessof the de Rham complex associated to a variation of Hodge structure, which is proved by
the infinitesimal method in Hodge theory.
References[1] J.-L. Colliot-Thélène, L’arithmétique des zéro-cycles (exposé aux Journées arithmétiques deBordeaux, 1993), J. Théor. Nombres Bordeaux 7 (1995), 51–73.
[2] M. Asakura and S. Saito: Surfaces over a p-adic field with infinite torsion in the Chow groupof 0-cycles, to appear in Algebra and Number Theory.
[3] S. Saito and K. Sato: A finiteness theorem on zero-cycles over p-adic fields, to appear in Ann.
of Math.