Class field theory for open curves over local fields

Toshiro Hiranouchi

doi:10.5802/jtnb.1036

Toshiro Hiranouchi ¹

¹ Department of Basic Sciences, Graduate School of Engineering, Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi, Fukuoka, 804-8550, Japan

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

Résumé
Abstract

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.

Reçu le : 2016-10-11
Révisé le : 2017-02-22
Accepté le : 2017-05-11
Publié le : 2018-12-06

MR Zbl

DOI : 10.5802/jtnb.1036

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

Affiliations des auteurs :

Toshiro Hiranouchi ¹

¹ Department of Basic Sciences, Graduate School of Engineering, Kyushu Institute of Technology 1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi, Fukuoka, 804-8550, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2018__30_2_501_0,
     author = {Toshiro Hiranouchi},
     title = {Class field theory for open curves over local fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {501--524},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1036},
     zbl = {07081559},
     mrnumber = {3891324},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1036/}
}

TY  - JOUR
AU  - Toshiro Hiranouchi
TI  - Class field theory for open curves over local fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 501
EP  - 524
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1036/
DO  - 10.5802/jtnb.1036
LA  - en
ID  - JTNB_2018__30_2_501_0
ER  -

%0 Journal Article
%A Toshiro Hiranouchi
%T Class field theory for open curves over local fields
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 501-524
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1036/
%R 10.5802/jtnb.1036
%G en
%F JTNB_2018__30_2_501_0

Toshiro Hiranouchi. Class field theory for open curves over local fields. Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 501-524. doi : 10.5802/jtnb.1036. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1036/

Bibliographie
Cité par

[1] Ahmed Abbes; Takeshi Saito Ramification of local fields with imperfect residue fields, Am. J. Math., Volume 124 (2002) no. 5, pp. 879-920 | DOI | MR | Zbl

[2] Ahmed Abbes; Takeshi Saito Analyse micro-locale $l$ -adique en caractéristique $p > 0$ : le cas d’un trait, Publ. Res. Inst. Math. Sci., Volume 45 (2009) no. 1, pp. 25-74 | DOI | Zbl

[3] Michael Artin; Alexander Grothendieck; Jean-Louis Verdier Theorie de topos et cohomologie etale des schemas I, II, III (SGA 4), Lecture Notes in Mathematics, 269, 370, 305, Springer, 1972-1973 | Zbl

[4] Pierre Deligne Cohomologie étale (SGA 4 $\frac{1}{2}$ ), Lecture Notes in Mathematics, 569, Springer, 1977, iv+312 pages | Zbl

[5] Ivan B. Fesenko Topological Milnor $K$ -groups of higher local fields, Invitation to higher local fields (Münster, 1999) (Geometry and Topology Monographs), Volume 3, Geometry and Topology Publications, 2000, pp. 61-74 | MR | Zbl

[6] Ivan B. Fesenko Sequential topologies and quotients of Milnor $K$ -groups of higher local fields, Algebra Anal., Volume 13 (2001) no. 3, pp. 198-221 | MR

[7] Ivan B. Fesenko; Sergei V. Vostokov Local fields and their extensions, Translations of Mathematical Monographs, 121, American Mathematical Society, 2002 | MR | Zbl

[8] Patrick Forré The kernel of the reciprocity map of varieties over local fields, J. Reine Angew. Math., Volume 698 (2015), pp. 55-69 | MR | Zbl

[9] Lei Fu Etale cohomology theory, Nankai Tracts in Mathematics, 13, World Scientific, 2011, ix+611 pages | MR | Zbl

[10] Revêtements étales et groupe fondamental, Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1) (Alexander Grothendieck, ed.), Lecture Notes in Mathematics, 224, Springer, 1971, xxii+447 pages | Zbl

[11] Toshiro Hiranouchi Class field theory for open curves over $p$ -adic fields, Math. Z., Volume 266 (2010) no. 1, pp. 107-113 | DOI | MR | Zbl

[12] Roland Huber étale cohomology of Henselian rings and cohomology of abstract Riemann surfaces of fields, Math. Ann., Volume 295 (1993) no. 4, pp. 703-708 | DOI | MR | Zbl

[13] Luc Illusie Complexe de de Rham-Witt (Astérisque), Volume 63, Société Mathématique de France, 1978, pp. 83-112 | Zbl

[14] Uwe Jannsen; Shuji Saito Kato homology of arithmetic schemes and higher class field theory over local fields, Doc. Math. (2003), pp. 479-538 | MR | Zbl

[15] Kazuya Kato A generalization of local class field theory by using $K$ -groups. I, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 26 (1979) no. 2, pp. 303-376 | MR | Zbl

[16] Kazuya Kato Galois cohomology of complete discrete valuation fields, Algebraic $K$ -theory, Part II (Oberwolfach, 1980) (Lecture Notes in Mathematics), Volume 967, Springer, 1980, pp. 215-238 | Zbl

[17] Kazuya Kato A generalization of local class field theory by using $K$ -groups. II, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 27 (1980) no. 3, pp. 603-683 | MR | Zbl

[18] Kazuya Kato Swan conductors for characters of degree one in the imperfect residue field case, Algebraic $K$ -theory and algebraic number theory (Honolulu, 1987) (Contemporary Mathematics), Volume 83, American Mathematical Society, 1987, pp. 101-131 | DOI | Zbl

[19] Kazuya Kato; Shuji Saito Two-dimensional class field theory, Galois groups and their representations (Nagoya, 1981) (Advanced Studies in Pure Mathematics), Volume 2, North-Holland, 1981, pp. 103-152 | Zbl

[20] Kazuya Kato; Shuji Saito Global class field theory of arithmetic schemes, Applications of algebraic K-theory to algebraic geometry and number theory (Boulder, 1983) (Contemporary Mathematics), Volume 55, American Mathematical Society, 1983, pp. 255-331 | Zbl

[21] Aleksandr S. Merkurʼev; Andreĭ A. Suslin $K$ -cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat., Volume 46 (1982) no. 5, pp. 1011-1046 | MR | Zbl

[22] James S. Milne Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980, xiii+323 pages | MR | Zbl

[23] James S. Milne Arithmetic duality theorems, BookSurge, 2006, viii+339 pages | Zbl

[24] John W. Milnor Algebraic $K$ -theory and quadratic forms, Invent. Math., Volume 9 (1970), pp. 318-344 | DOI | MR | Zbl

[25] John W. Milnor Introduction to algebraic $K$ -theory, Annals of Mathematics Studies, 72, Princeton University Press, 1971, xiii+184 pages | MR | Zbl

[26] Jürgen Neukirch Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Springer, 1999, xvii+571 pages | Zbl

[27] Jürgen Neukirch; Alexander Schmidt; Kay Wingberg Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008, xv+825 pages | MR | Zbl

[28] Wayne Raskind Abelian class field theory of arithmetic schemes, $K$ -theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, 1992) (Proceedings of Symposia in Pure Mathematics), Volume 58, American Mathematical Society, 1992, pp. 85-187 | MR | Zbl

[29] Shuji Saito Class field theory for curves over local fields, J. Number Theory, Volume 21 (1985) no. 1, pp. 44-80 | DOI | MR | Zbl

[30] Shuji Saito A global duality theorem for varieties over global fields, Algebraic $K$ -theory: connections with geometry and topology (Lake Louise, 1987) (NATO ASI Series, Series C: Mathematical and Physical Sciences), Volume 279, Kluwer Academic Publishers, 1987, pp. 425-444 | Zbl

[31] Jean-Pierre Serre Corps locaux, Publications de l’Université de Nancago, VIII, Hermann, 1980 | Zbl

[32] Takao Yamazaki Class field theory for a product of curves over a local field, Math. Z., Volume 261 (2009) no. 1, pp. 109-121 | DOI | MR | Zbl

[33] Takao Yamazaki The Brauer-Manin pairing, class field theory, and motivic homology, Nagoya Math. J., Volume 210 (2013), pp. 29-58 | DOI | MR | Zbl

[34] Teruyoshi Yoshida Finiteness theorems in the class field theory of varieties over local fields, J. Number Theory, Volume 101 (2003) no. 1, pp. 138-150 | DOI | MR | Zbl

Cité par Sources :