Dans cette courte note, nous donnons une nouvelle approche pour prouver la modularité des représentations galoisiennes -adiques en utilisant une méthode d’approximations -adiques. Cela englobe quelques uns des résultats bien connus de Wiles et Taylor dans de nombreux cas mais pas tous. Une caractéristique de cette nouvelle approche est qu’elle travaille directement avec la représentation galoisienne -adique dont on cherche à établir la modularité. Les trois ingrédients essentiels sont une technique de cohomologie galoisienne de Ramakrishna, un résultat de montée de niveau de Ribet, Diamond, Taylor et une version mod du principe de descente de niveau de Mazur.
In this short note we give a new approach to proving modularity of -adic Galois representations using a method of -adic approximations. This recovers some of the well-known results of Wiles and Taylor in many, but not all, cases. A feature of the new approach is that it works directly with the -adic Galois representation whose modularity is sought to be established. The three main ingredients are a Galois cohomology technique of Ramakrishna, a level raising result due to Ribet, Diamond, Taylor, and a mod version of Mazur’s principle for level lowering.
@article{JTNB_2004__16_1_179_0,
author = {Khare, Chandrashekhar},
title = {Modularity of $p$-adic {Galois} representations via $p$-adic approximations},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {179--185},
publisher = {Universit\'e Bordeaux 1},
volume = {16},
number = {1},
year = {2004},
doi = {10.5802/jtnb.440},
zbl = {1082.11030},
mrnumber = {2145579},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.440/}
}
TY - JOUR AU - Khare, Chandrashekhar TI - Modularity of $p$-adic Galois representations via $p$-adic approximations JO - Journal de théorie des nombres de Bordeaux PY - 2004 SP - 179 EP - 185 VL - 16 IS - 1 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.440/ DO - 10.5802/jtnb.440 LA - en ID - JTNB_2004__16_1_179_0 ER -
%0 Journal Article %A Khare, Chandrashekhar %T Modularity of $p$-adic Galois representations via $p$-adic approximations %J Journal de théorie des nombres de Bordeaux %D 2004 %P 179-185 %V 16 %N 1 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.440/ %R 10.5802/jtnb.440 %G en %F JTNB_2004__16_1_179_0
Khare, Chandrashekhar. Modularity of $p$-adic Galois representations via $p$-adic approximations. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 179-185. doi : 10.5802/jtnb.440. http://www.numdam.org/articles/10.5802/jtnb.440/
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