Non Trival Extensions of p-adic Galois Representations that are Trival at p.
CMI/MSRI Workshop: Modular Forms and Arithmetic
July 01, 2008 02:00 PM to 03:00 PM
Speakers:
Bellaiche, Joel
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Abstract: |
The Bloch-Kato Selmer group of a p-adic Galois representation of a number field is a space of extensions that are unramifed at almost all places, and that satisfies some Fontaine-theoretic properties at every places of K dividing p. The Bloch-Kato conjecture predicts the dimensions of those spaces, and some progresses have been made toward this conjecture, even if the general case seems still very far.
If we replace the Fontaine-Mazur conditions at some places dividing p by the condition of being trivial (or dually, by no condition at all) at those places, we obtain spaces that are often even more mysterious than the Bloch-Kato Selmer groups. Indeed, even in the simplest case of the trivial representation, knowing the dimension of those spaces is equivalent to a generalization of the Leopoldt conjecture, in the spirit of the one recently proposed by Calegari-Mazur, and in general even a conjectural formula for the dimension of those spaces seems unknown (only some special cases being in the scope of Jannsen's conjecture)
In my talk, I will discuss those issues and explain how one can construct, using generalizations of ideas of Ribet, together with the work of Chenevier and myself on the geometry of eigenvarieties, non-trivial elements in those spaces. |
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