
School of Mathematics and Statistics
Sigrid Wortmann
Universität Heidelberg
On Galois representations of motives.
Friday 24th September, 121pm, Carslaw 375.
Let X be a smooth projective variety defined over a number
field k. For every prime l one has
ladic cohomology groups
H^{i}_{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_{l}^{i}(X)(r) )_{l}
of ladic 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
H^{i}_{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 ladic
representations in this situation.
