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Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture

Luis Dieulefait
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Abstract

In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate $λ$-adic 2-dimensional Galois representation unramified at 3 and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p^2 \}) = \Q $ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of conductor 1 and apply this to the irreducibility of Galois representations on level 1 genus 2 Siegel cusp forms.

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Luis Dieulefait. 2003-04-27. Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture. https://arxiv.org/abs/math/0304433

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