Geodesic
Lifting Galois representations and a conjecture of Fontaine and Mazur
Documenta mathematica, Tome 6 (2001), pp. 419-445.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Mumford has constructed 4-dimensional abelian varieties with trivial endomorphism ring, but whose Mumford--Tate group is much smaller than the full symplectic group. We consider such an abelian variety, defined over a number field $F$, and study the associated $p$-adic Galois representation. For $F$ sufficiently large, this representation can be lifted to $\mathbf{G}_m(\mathbf{Q}_p)\times\mathrm{SL}_2(\mathbf{Q}_p)^3$. Such liftings can be used to construct Galois representations which are geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question predicts that these representations should come from algebraic geometry. We confirm the conjecture for the representations constructed here.
Classification : 11G10, 11F80, 14K15
Keywords: geometric Galois representation, fontaine--Mazur conjecture, motives 
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     author = {Noot, Rutger},
     title = {Lifting {Galois} representations and a conjecture of {Fontaine} and {Mazur}},
     journal = {Documenta mathematica},
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     volume = {6},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2001__6__a5/}
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Noot, Rutger. Lifting Galois representations and a conjecture of Fontaine and Mazur. Documenta mathematica, Tome 6 (2001), pp. 419-445. http://geodesic.mathdoc.fr/item/DOCMA_2001__6__a5/