On a conjecture of Bloch and Kato on local motivic $\ell$-adic
representations
(Un exposé de Uwe Jannsen au séminaire de
théorie des nombres de Chevaleret le 13 déembre 2005)
Résumé :
In their seminal paper on the Tamagawa number conjecture,
Bloch and Kato proposed a certain conjecture on the $\ell$-adic
realizations of motives over number fields. It just concerns the
action of the local Galois group $G_v$ at a place $v$ and is only
a question if the reduction at $v$ is bad. The conjecture is that
the adelic representation contains a $\hat Z$-lattice
$T$ such that the torsion of $T_{G_v}$ is finite. In joint work
with Petra Seidel, this is be proved for (the tensor category
generated by) curves, abelian varieties and surfaces. In fact,
we are led to a finer conjecture, in terms of the monodromy
filtration, which has better functorial properties for the tensor
product, and are able to prove this stronger conjecture in the
mentioned cases.