Sydney University Algebra Seminar

# Algebra Seminar

### On Galois representations of motives.

Friday 24th September, 12-1pm, Carslaw 375.

Let X be a smooth projective variety defined over a number field k. For every prime l one has l-adic cohomology groups Hil(X)(r) (0\le i\le 2\dim(X), r\in Z). They have a natural action of the absolute Galois group Gk of k and thus define systems (rhol: Gk--> AutQl (Hli(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 rhol: Gk-->AutZl(Tl(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 (\phol)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 rhol: Gk--> AutQl (Vl) where Vl is a direct factor of representations Hil(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.