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Deforming semistable Galois representations.

J M Fontaine1

  • 1Université de Paris-Sud, Mathématique, Bâtiment 425, F-91405 Orsay Cedex, France.

Proceedings of the National Academy of Sciences of the United States of America
|October 19, 2001
PubMed
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This study explores extending Wiles's proof methods for Fermat's Last Theorem (FLT) by examining conditions for p-adic representations of the absolute Galois group of rational numbers (Gal(Q/Q)). The research focuses on local conditions at prime p to ensure related representations also derive from automorphic forms.

Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Representation Theory

Background:

  • Wiles's proof of Fermat's Last Theorem (FLT) relies on the connection between p-adic representations and automorphic forms.
  • A key idea is that 'nearby' representations also correspond to automorphic forms.

Purpose of the Study:

  • To investigate the local conditions at prime p necessary to extend Wiles's methodology.
  • To determine criteria for a p-adic representation V' to be associated with an automorphic form, given a similar representation V.

Main Methods:

  • Analysis of p-adic representations of the absolute Galois group of rational numbers (Gal(Q/Q)).
  • Exploration of the 'closed enough' condition between representations V and V' in the context of Wiles's proof.

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Main Results:

  • Identifies the crucial role of local conditions at prime p in extending Wiles's arguments.
  • Suggests that specific local properties at p are essential for transferring the automorphic property between related representations.

Conclusions:

  • Extending Wiles's methods requires careful consideration of local behavior at prime p.
  • The findings provide insights into the structure of Galois representations and their connection to automorphic forms.