### On an analogue for number fields of a conjecture of de Jong on F_q[[t]]-analytic extensions of function fields, by Gebhard Boeckle

We formulate a finiteness conjecture on the image of the absolute
Galois group of totally real fields under a linear representation
over a local field of finite characteristic. This
parallels a recent conjecture of de Jong in the function field case.
We give some group-theoretical motivation, some evidence for the conjecture
and explain its relation to properties of universal deformation spaces
of even residual representations.

Gebhard Boeckle <boeckle@math.ethz.ch>