Class field theory for open curves over local fields

Abstract

We investigate the class field theory for products of open curves over a local field. In particular, we determine the kernel of the reciprocity homomorphism.Since the middle vertical map is surjective, the assertion follows from the above diagram.

Higher Ideles and Class Field Theory

“…Note that the generic fiber X η is a smooth variety over the local field K. Its class field theory has been studied in several cases, for example the case d = 1 is well understood by work of Bloch and Saito, see [Sai85] and [Hir16]. In [For15], Forré determines the kernel of the reciprocity map in unramified ℓ-adic class field theory in the higher dimension case.…”