On the p-adic Langlands correspondence for algebraic tori
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 133-158.

We extend the results by R.P. Langlands on representations of (connected) abelian algebraic groups. This is done by considering characters into any divisible abelian topological group. With this we can then prove what is known as the abelian case of the p-adic Langlands program.

Nous étendons les résultats de R.P. Langlands sur les représentations des groupes algébriques abéliens connexes. Pour démontrer nos théorèmes, nous considérons les caractères à valeurs dans un groupe topologique abélien divisible quelconque. Cela nous permet de prouver le cas abélien du programme de Langlands p-adique.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1114
Classification: 11R39
Keywords: p-adic, Langlands, tori
Christopher Birkbeck 1

1 Department of Mathematics University College London Gower street London, WC1E 6BT, UK
@article{JTNB_2020__32_1_133_0,
     author = {Christopher Birkbeck},
     title = {On the $p$-adic {Langlands} correspondence for algebraic tori},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {133--158},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1114},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1114/}
}
TY  - JOUR
TI  - On the $p$-adic Langlands correspondence for algebraic tori
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2020
DA  - 2020///
SP  - 133
EP  - 158
VL  - 32
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1114/
UR  - https://doi.org/10.5802/jtnb.1114
DO  - 10.5802/jtnb.1114
LA  - en
ID  - JTNB_2020__32_1_133_0
ER  - 
%0 Journal Article
%T On the $p$-adic Langlands correspondence for algebraic tori
%J Journal de Théorie des Nombres de Bordeaux
%D 2020
%P 133-158
%V 32
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1114
%R 10.5802/jtnb.1114
%G en
%F JTNB_2020__32_1_133_0
Christopher Birkbeck. On the $p$-adic Langlands correspondence for algebraic tori. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 133-158. doi : 10.5802/jtnb.1114. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1114/

[1] Emil Artin; John T. Tate Class field theory, AMS Chelsea Publishing, 2009 | Zbl: 1179.11040

[2] Christophe Breuil The emerging p-adic Langlands programme, Proceedings of the international congress of mathematicians (ICM 2010). Vol. II: Invited lectures (2010), pp. 203-230 | MR: 2827792 | Zbl: 1368.11123

[3] Kenneth S. Brown Cohomology of Groups, Graduate Texts in Mathematics, Volume 87, Springer, 1982 | Zbl: 0584.20036

[4] Robert P. Langlands Representations of abelian algebraic groups, Pac. J. Math. (1997) no. Special Issue, pp. 231-250 (Olga Taussky-Todd: in memoriam) | Article | MR: 1610871 | Zbl: 0910.11045

[5] John S. Milne Arithmetic duality theorems, BookSurge, 2006 | Zbl: 1127.14001

[6] Jürgen Neukirch Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, Volume 322, Springer, 1999 (transl. by N. Schappacher) | Zbl: 0956.11021

[7] Joseph J. Rotman An Introduction to Homological Algebra, Universitext, Springer, 2009 | Zbl: 1157.18001

[8] Jean-Pierre Serre Local fields, Graduate Texts in Mathematics, Springer, 1980 (transl. by M. J. Greenberg) | Zbl: 0423.12017

[9] John T. Tate Number theoretic background, Automorphic forms, representations and L-functions (Proceedings of Symposia in Pure Mathematics) Volume 33, Part 2 (1979), pp. 3-26 | Article | Zbl: 0422.12007

[10] Charles A. Weibel An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Volume 38, Cambridge University Press, 1995 | Zbl: 0834.18001

[11] Edwin Weiss Cohomology of Groups, Pure and Applied Mathematics, Volume 34, Academic Press Inc., 1969 | MR: 263900 | Zbl: 0192.34204

Cited by Sources: