Modularity of $p$-adic Galois representations via $p$-adic approximations

Khare, Chandrashekhar

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 179-185.

Dans cette courte note, nous donnons une nouvelle approche pour prouver la modularité des représentations galoisiennes $p$-adiques en utilisant une méthode d’approximations $p$-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 $p$-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 $p^n$ du principe de descente de niveau de Mazur.

In this short note we give a new approach to proving modularity of $p$-adic Galois representations using a method of $p$-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 $p$-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 $p^n$ version of Mazur’s principle for level lowering.

[DT] F. Diamond, R. Taylor, Lifting modular mod $l$ representations. Duke Math. J. 74 no. 2 (1994), 253–269. | MR | Zbl

[K] C. Khare, On isomorphisms between deformation rings and Hecke rings. To appear in Inventiones mathematicae, preprint available at http://www.math.utah.edu/~shekhar/papers.html | MR | Zbl

[K1] C. Khare, Limits of residually irreducible $p$-adic Galois representations. Proc. Amer. Math. Soc. 131 (2003), 1999–2006. | MR | Zbl

[KR] C. Khare, R. Ramakrishna, Finiteness of Selmer groups and deformation rings. To appear in Inventiones mathematicae, preprint available at http://www.math.utah.edu/~shekhar/papers.html | MR | Zbl

[Ri] K. Ribet, Report on mod $\ell$ representations of $Gal(\overline{\mathbf{Q}}/\mathbf{Q})$. In Motives, Proc. Sympos. Pure Math. 55, part 2 (1994), 639–676. | MR | Zbl

[R] R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur. Annals of Math. 156 (2002), 115–154. | MR | Zbl

[T] R. Taylor, On icosahedral Artin representations II. To appear in American J. of Math. | MR | Zbl

[TW] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), 553–572. | MR | Zbl

[W] A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. 141 (1995), 443–551. | MR | Zbl