Algebra Seminar

Let X be a smooth projective variety defined over a number field k. For every prime l one has l-adic cohomology groups Hⁱ_l(X)(r) (0\le i\le 2\dim(X), r\in Z). They have a natural action of the absolute Galois group G_k of k and thus define systems (rho_l: G_k--> Aut_Ql (H_lⁱ(X)(r) )_l of l-adic representations. They are the subject of several conjectures, but not much is provable in general. However, if X is an elliptic curve without complex multiplication Serre showed that rho_l: G_k-->Aut_Zl(T_l(X)) is surjective for almost all l. This result was extended by Ribet and Serre to certain abelian varieties with complex multiplication. The case of abelian varieties with complex multiplication is easier because the the system (\pho_l)_l is induced by some algebraic Hecke characters of some finite extension of k. In our talk we are going to determine (in special situations) the image of the representations rho_l: G_k--> Aut_Ql (V_l) where V_l is a direct factor of representations Hⁱ_l(X)(r), (X smooth projective) in a compatible (i.e. motivic) way. Assuming that our representations are not induced by an algebraic Hecke character we can show that the image is ``as big as possible'' and moreover confirm several conjectures on l-adic representations in this situation.