Surfaces over a p-adic field with infinite torsion in the Chow group of 0-cycles

Quintic surface over 𝑝-adic local fields with infinite 𝑝-primary torsion in the Chow group of 0-cycles

“…Our proof is comparable with that of [1], however a new difficulty appears in case l = p. Let us recall the outline of the proof of [1] briefly. It follows from the universal coefficient theorem on Bloch's higher Chow group that we have the exact sequence 0 −→ CH 2 (X, 1) ⊗ Q l /Z l i −→ CH 2 (X, 1; Q l /Z l ) −→ CH 2 (X)[l ∞ ] −→ 0 (1.0.1) for any l (possibly l = p).…”

“…It follows from the universal coefficient theorem on Bloch's higher Chow group that we have the exact sequence 0 −→ CH 2 (X, 1) ⊗ Q l /Z l i −→ CH 2 (X, 1; Q l /Z l ) −→ CH 2 (X)[l ∞ ] −→ 0 (1.0.1) for any l (possibly l = p). The proof of [1] breaks up into two steps. We first showed that if X is generic then CH 2 (X, 1) ⊗ Q l /Z l contains only decomposable elements supported on hyperplane section (cf.…”

“…In the paper [1] S. Saito and the author constructed a surface X over a p-adic local field such that the l-primary torsion part CH 0 (X)[l ∞ ] of the Chow group of 0-cycles is infinite for l = p, which gave a counter-example to a folklore conjecture. The purpose of this paper is to show that there is such an example even for l = p: Theorem 1.0.1 (Theorem 4.1.2) There is a quintic surface X ⊂ P 3 Qp over Q p such that the p-primary torsion part CH 0 (X × Qp K)[p ∞ ] is not finite for arbitrary finite extension K of Q p .…”

A restriction isomorphism for zero-cycles with coefficients in Milnor K-theory

“…(1) be the generic point of a lift Z 1 of x 0 which intersects X 0 transversally in x 0 . We may now assume that α = (α y , α z ) ∈ ker(res : 1) . This follows from the fact that for every z ∈ X g (1) which intersects X 0 in x 0 , we can lift res(α z ) to an element α y ∈ K M j−d k(y) such that ∂(α y ) = ∂(α z ).…”

Higher Chow cycles on Abelian surfaces and a non-Archimedean analogue of the Hodge--conjecture

“…It is still expected to be true if X is defined over a number field. In fact Asakura and Saito [AS07] show that for certain generic surfaces a non-Archimedean version of this is false over a p-adic field as well. In this case too, however, one expects the conjecture to be true for varieties defined over global fields.…”