Non-divisible cycles on surfaces over local fields Original Research Article

In this paper, we are concerned with the reciprocity map of unramified class field theory for smooth projective surfaces over non-archimedean local fields which do not have potentially good reduction. We will construct two types of smooth projective surfaces whose reciprocity maps modulo positive integers are not injective. The first type is the case where the kernel of the reciprocity map is not divisible. The second is the case where the kernel of the reciprocity map is divisible, but where nevertheless the reciprocity map modulo some integer is not injective.