Lifting systems of Galois representations associated to abelian varieties

Details Lifting systems of Galois representations associated to abelian varieties Lifting systems of Galois representations associated to abelian varieties

2004-07-21

arxiv.org/abs/math/0407360v3

Journal of the Ramanujan Mathematical Society, 21 No. 4 (2006), pp. 299-342

Mathematics

Number Theory

50 pages. In version 3 some clarifications and examples are added, some proofs streamlined. Except for the layout, this versio

Scientific paper

This paper treats what we call `weak geometric liftings' of Galois representations associated to abelian varieties. This notion can be seen as a generalization of the idea of lifting a Galois representation along an isogeny of algebraic groups. The weaker notion only takes into account an isogeny of the derived groups and disregards the centres of the groups in question. The weakly lifted representations are required to be geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question states that any irreducible geometric representation is a twist of a subquotient of an \'etale cohomology group of an algebraic variety over a number field. It is shown that a Galois representation associated to an abelian variety admits a weak geometric lift to a group with simply connected derived group. In certain cases, such a weak geometric lift is itself associated to an abelian variety. This means that the conjecture of Fontaine and Mazur is confirmed for these representations. In other cases, one may find a lift which can not be found back in the etale cohomology of any abelian variety. The Fontaine--Mazur conjecture remains open for these representations. Nevertheless, certain consequences of the conjecture can be established.

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