Constructing Galois Representations with Very Large Image

http://dx.doi.org/10.4153/CJM-2008-009-7
Canad. J. Math. 60(2008), 208-221

  Published:2008-02-01
 Printed: Feb 2008

Starting with a 2-dimensional mod $p$ Galois representation, we construct a deformation to a power series ring in infinitely many variables over the $p$-adics. The image of this representation is full in the sense that it contains $\SL_2$ of this power series ring. Furthermore, all ${\mathbb Z}_p$ specializations of this deformation are potentially semistable at $p$.