Resume "Galois representations ..." (Rutger Noot)

Abelian varieties - Galois representations and properties of ordinary reduction

Compositio Mathematica 97 (1995), p. 161-171.

Summary/Résumé

In this paper, two results are proven on the Galois representations associated to Abelian varieties. The first one states the following. Let l be a prime number and suppose that G is the Mumford-Tate group ofan Abelian variety X/C. Then there exist a number field F and an Abelian variety Y/F with Mumford-Tate group G, such that the image of the l-adic Galois representation associated to Y is an open and Zariski dense subgroup of G(Q_l).

To state the second result, let k>0 be an integer. There exists an Abelian variety X over C such that the Mumford-Tate group G of X is geometrically isogenous to a product of 2k+1 copies of the symplectic group Sp(2) and one copy of the multiplicative group. The first theorem therefore implies that there exist a number field F and an Abelian variety X/F such that the image of the associated l-adic Galois representation is open in the group G(Q_l). For any such Abelian variety X/F, the theorem states that there exist a finite extension F' of F and a set P of places of F', of Dirichlet density 1, such that X has good and ordinary reduction at every place v in P.

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Dernière modification: le 3 septembre 2003