Class field theory for open curves over p-adic fields

Abstract

We introduce the idèle class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends Saito's class field theory for projective curves (Saito in J Number Theory 21: 1985).

The Brauer–Manin pairing, class field theory, and motivic homology

“…In Section 5, we consider the case where X is a curve. In this case, the map ψ X was already studied by Scheiderer and van Hamel [33] (see Theorem 5.1 below), and the map ρ X is closely related to work of Hiranouchi [16] (see Theorem 5.4 and Remark 5.5 below). When X is of dimension 2 or higher, the maps ψ X and ρ X are not very close to an isomorphism even if X is projective over k. We study several examples of surfaces in Section 6.…”

The Brauer–Manin pairing, class field theory, and motivic homology

“…In this case, the map ψ X was already studied by Scheiderer-van Hamel [34] (see Theorem 5.1 below), and the map ρ X is closely related to work of Hiranouchi [16] (see Theorem 5.4 and Remark 5.5 below). When X is of dimension two or higher, the maps ψ X and ρ X are not very close to an isomorphism even if X is projective over k. We study several examples of surfaces in §6.…”

Class field theory for open curves over local fields

“…In the case where X is a (smooth) curve, the assertion follows from Sect. 2 of [5]. The main ingredient of the discussion in op.…”

“…First, we introduce an abelian group C(X) which is called the idèle class group for X as in [5] (Def. 2.3), and the reciprocity map ρ X : C(X) → π ab 1 (X)…”

“…Here, a local field means a complete discrete valuation field with finite residue field. For a local field with characteristic 0, a large number of studies have been made even for higher dimensional open varieties over local fields (e.g., [13], [11], [31], and [32]). Accordingly, our main interest is in the case of positive characteristic local fields.…”