Abstract:
Let p>5 be a prime and C the complex numbers. In two recent articles, 
Rohrlich proved `rigidity' for the group PSL(2,Z_p[[T]]), obtained, using 
modular function fields , this group as a Galois group over C(t), and 
derived from this interesting consequences for Galois representations 
attached to the Tate modules of elliptic curves. Furthermore in an 
unpublished preprint, he established that the corresponding Galois 
representation of the absolute Galois group of C(t) is universal. 

The latter observation lead me to turn around Rohrlich's approach. 
So one first computes universal deformation rings for n-dimensional 
projective representations of the absolute Galois group of k(t), where k 
a separably closed field. If the residual representation is `geometrically 
rigid', which happens to be the case for typical surjective representation 
to PSL(2,F_p), then certain universal deformations will be `geometrically 
rigid', too.

On the one hand, this gives new proofs for most of the results of 
Rohrlich. On the other hand, by specializing t and T suitably, this 
yields p-adic Galois representation of the absolute Galois group of 
Q(\zeta) which surject onto the group SL(2,Z_p[\zeta+\zeta^{-1}]), where 
\zeta is any p^n-th root of unity. Similarly, one obtains p-adic Galois 
representations of function fields over finite fields which surject onto 
the same group, and which are are unramified after a finite base extension.