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.
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.
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1036
Mots-clés : Class field theory, local fields
Toshiro Hiranouchi 1

@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, Volume 30 (2018) no. 2, pp. 501-524. doi : 10.5802/jtnb.1036. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1036/
[1] Ramification of local fields with imperfect residue fields, Am. J. Math., Volume 124 (2002) no. 5, pp. 879-920 | DOI | MR | Zbl
[2] Analyse micro-locale -adique en caractéristique : le cas d’un trait, Publ. Res. Inst. Math. Sci., Volume 45 (2009) no. 1, pp. 25-74 | DOI | Zbl
[3] 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] Cohomologie étale (SGA 4), Lecture Notes in Mathematics, 569, Springer, 1977, iv+312 pages | Zbl
[5] Topological Milnor -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] Sequential topologies and quotients of Milnor -groups of higher local fields, Algebra Anal., Volume 13 (2001) no. 3, pp. 198-221 | MR
[7] Local fields and their extensions, Translations of Mathematical Monographs, 121, American Mathematical Society, 2002 | MR | Zbl
[8] The kernel of the reciprocity map of varieties over local fields, J. Reine Angew. Math., Volume 698 (2015), pp. 55-69 | MR | Zbl
[9] 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] Class field theory for open curves over -adic fields, Math. Z., Volume 266 (2010) no. 1, pp. 107-113 | DOI | MR | Zbl
[12] 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] Complexe de de Rham-Witt (Astérisque), Volume 63, Société Mathématique de France, 1978, pp. 83-112 | Zbl
[14] Kato homology of arithmetic schemes and higher class field theory over local fields, Doc. Math. (2003), pp. 479-538 | MR | Zbl
[15] A generalization of local class field theory by using -groups. I, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 26 (1979) no. 2, pp. 303-376 | MR | Zbl
[16] Galois cohomology of complete discrete valuation fields, Algebraic -theory, Part II (Oberwolfach, 1980) (Lecture Notes in Mathematics), Volume 967, Springer, 1980, pp. 215-238 | Zbl
[17] A generalization of local class field theory by using -groups. II, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 27 (1980) no. 3, pp. 603-683 | MR | Zbl
[18] Swan conductors for characters of degree one in the imperfect residue field case, Algebraic -theory and algebraic number theory (Honolulu, 1987) (Contemporary Mathematics), Volume 83, American Mathematical Society, 1987, pp. 101-131 | DOI | Zbl
[19] 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] 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] -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] Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980, xiii+323 pages | MR | Zbl
[23] Arithmetic duality theorems, BookSurge, 2006, viii+339 pages | Zbl
[24] Algebraic -theory and quadratic forms, Invent. Math., Volume 9 (1970), pp. 318-344 | DOI | MR | Zbl
[25] Introduction to algebraic -theory, Annals of Mathematics Studies, 72, Princeton University Press, 1971, xiii+184 pages | MR | Zbl
[26] Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Springer, 1999, xvii+571 pages | Zbl
[27] Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2008, xv+825 pages | MR | Zbl
[28] Abelian class field theory of arithmetic schemes, -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] Class field theory for curves over local fields, J. Number Theory, Volume 21 (1985) no. 1, pp. 44-80 | DOI | MR | Zbl
[30] A global duality theorem for varieties over global fields, Algebraic -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] Corps locaux, Publications de l’Université de Nancago, VIII, Hermann, 1980 | Zbl
[32] 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] The Brauer-Manin pairing, class field theory, and motivic homology, Nagoya Math. J., Volume 210 (2013), pp. 29-58 | DOI | MR | Zbl
[34] 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
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