## Local-global type conjectures on the image of Chow groups in étale cohomology. (Conjectures de type local-global sur l’image des groupes de Chow dans la cohomologie étale.)(French)Zbl 0981.14003

Raskind, Wayne (ed.) et al., Algebraic $$K$$-theory. Proceedings of an AMS-IMS-SIAM summer research conference, Seattle, WA, USA, July 13-24, 1997. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 67, 1-12 (1999).
Let $$X$$ be a smooth projective variety of dimension $$d$$ over a global field $$k$$. Assume that $$X$$ is geometrically integral. For such a variety, the author formulates the following conjecture:
Let $$i$$ be a positive integer and $$l$$ be a prime number different from the characteristic of $$k$$. Assume that for each place $$v$$ of $$k$$ a cycle $$z_v$$ is given, belonging to $$\text{CH}^i(X_v)$$, the Chow group of codimension $$i$$ cycles on $$X_v$$, modulo rational equivalence (here $$X_v$$ denotes $$X\times_k k_v$$). Assume also that every class in $$H^{2j}(X,\mathbb Q_l/\mathbb Z_l(j))$$ is orthogonal to the whole family $$\{z_v\}$$. Then, for all positive $$n$$, there exists a global cycle $$z_n\in \text{CH}^i(X)$$, such that the class of $$z_n$$ in $$\tilde H^{2i}(X_v, \mu^{\otimes i}_{l^n})$$ coincides with the class of $$z_v$$ for all $$v$$.
The author then proves the following two facts, as a consequence of a general proposition:
Proposition 3.1: (i) the proposed conjecture is true if $$k$$ is a number field, the codimension $$i$$ is one and the Tate-Shafarevich group of the Picard variety of $$X$$ is a finite group;
(ii) if $$k$$ is the function field of a smooth projective curve $$C$$ over a finite field and $$X/k$$ admits a regular model $$\mathcal X /C$$, then the conjecture for $$\text{CH}^i(X)$$ follows from a suitable form of the Tate conjecture for the Chow groups on $$\mathcal X$$.
For the entire collection see [Zbl 0931.00031].

### MSC:

 14C15 (Equivariant) Chow groups and rings; motives 14F20 Étale and other Grothendieck topologies and (co)homologies

### Keywords:

Tate conjecture for the Chow groups