Finiteness Conjectures for Fl[[T]]-Analytic Extensions of Number Fields

Abstract

We formulate a finiteness conjecture on the image of the absolute Galois group of totally real fields under an even linear representation over a local field of positive characteristic. This is motivated by a recent conjecture of de Jong in the function field case. We discuss the relation to some conjectures of Boston which arise from the conjectures of Fontaine and Mazur and give group-theoretical reformulations of our conjectures. As we will explain, our conjectures have consequences for the structure of universal deformation rings of even residual representations. Finally, we give some evidence for the conjectures themselves and for their consequences.