Class field theory for a product of curves over a local field

Abstract

We prove that the kernel of the reciprocity map for a product of curves over a p-adic field with split semi-stable reduction is divisible. We also consider the K 1 of a product of curves over a number field.

Class field theory for open curves over local fields

“…First, we recall the definition of the Mackey products and that of the Somekawa Kgroups following [12], [16] and [8]: Recall that a Mackey functor A over a field F is a contravariant functor from the category of étale schemes over F to that of abelian groups equipped with a covariant structure for finite morphisms satisfying some conditions (for the precise definition, see [12] Section 3, or [8], Sect. 2).…”

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

“…When X is proper over k, Manin [23], Bloch [1], and Saito [29] observed thatψ X andρ X factor, respectively, through CH 0 (X) := coker y∈X (1) k(y) * → Z 0 (X) , SK 1 (X) := coker y∈X (1) K 2 k(y) → Z 1 0 (X) ; the induced pairing CH 0 (X) × Br(X) → Q/Z and the induced map SK 1 (X) → π ab 1 (X) are called the Brauer-Manin pairing and reciprocity map of class field theory, respectively. Both are studied intensively by several authors (for the Brauer-Manin pairing, see [4]- [6], [28], [31], and [46]; for the reciprocity map, see [18], [19], [32], [40], [41], and [47]).…”

A filtration on the higher Chow group of zero cycles on an abelian variety

“…In Theorem 5.2 (i), we obtain an analogue in the higher case by using a result of Yamazaki [Y09,Lemma 2.4,Proposition 3.1]. If A has potentially good reduction or split semi-abelian reduction and s > 0, the quotient…”