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

Chandrashekhar Khare

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 179-185
  • ISSN: 1246-7405

Abstract

top
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.

How to cite

top

Khare, Chandrashekhar. "Modularity of $p$-adic Galois representations via $p$-adic approximations." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 179-185. <http://eudml.org/doc/249262>.

@article{Khare2004,
abstract = {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.},
author = {Khare, Chandrashekhar},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Galois representations; modularity; -adic approximation},
language = {eng},
number = {1},
pages = {179-185},
publisher = {Université Bordeaux 1},
title = {Modularity of $p$-adic Galois representations via $p$-adic approximations},
url = {http://eudml.org/doc/249262},
volume = {16},
year = {2004},
}

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
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 179
EP - 185
AB - 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.
LA - eng
KW - Galois representations; modularity; -adic approximation
UR - http://eudml.org/doc/249262
ER -

References

top
  1. F. Diamond, R. Taylor, Lifting modular mod l representations. Duke Math. J. 74 no. 2 (1994), 253–269. Zbl0809.11025MR1272977
  2. 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 Zbl1042.11031MR2004460
  3. C. Khare, Limits of residually irreducible p -adic Galois representations. Proc. Amer. Math. Soc. 131 (2003), 1999–2006. Zbl1150.11437MR1963742
  4. 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 Zbl1104.11026MR2004459
  5. K. Ribet, Report on mod ℓ representations of Gal ( Q ¯ / Q ) . In Motives, Proc. Sympos. Pure Math. 55, part 2 (1994), 639–676. Zbl0822.11034MR1265566
  6. R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur. Annals of Math. 156 (2002), 115–154. Zbl1076.11035MR1935843
  7. R. Taylor, On icosahedral Artin representations II. To appear in American J. of Math. Zbl1031.11031MR1981033
  8. R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141 (1995), 553–572. Zbl0823.11030MR1333036
  9. A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. 141 (1995), 443–551. Zbl0823.11029MR1333035

NotesEmbed ?

top

You must be logged in to post comments.