Class field theory for open curves over local fields

Hiranouchi, Toshiro

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 501-524.

Nous étudions la théorie des corps de classes des courbes ouvertes sur un corps local. Après avoir introduit l’application de réciprocité nous déterminons son noyau et son conoyau. La duale de Pontrjagin de l’application de réciprocitIé est également étudiée. Cela nous donne, sous certaines hypothèses, une correspondance bijective entre l’ensemble des revêtements étales abéliens et l’ensemble des sous-groupes ouverts d’indice fini du groupe des classes d’idèles.

We study the class field theory for open curves over a local field. After introducing the reciprocity map, we determine the kernel and the cokernel of this map. In addition to this, the Pontrjagin dual of the reciprocity map is also investigated. This gives the one to one correspondence between the set of abelian étale coverings and the set of finite index open subgroups of the idèle class group as in the classical class field theory under some assumptions.

Classification : 11R37, 11R58
Mots clés : Class field theory, local fields

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