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On the cycle class map for zero-cycles over local fields

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On the cycle class map for zero-cycles over local fields
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The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Hélène Esnault.
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