Deformations and the rigidity method

AbstractIn [D. Rohrlich, False division towers of elliptic curves, J. Algebra 229 (1) (2000) 249–279; D. Rohrlich, A deformation of the Tate module, J. Algebra 229 (1) (2000) 280–313], Rohrlich proved rigidity for PSL2(Zp〚T〛) for p>5, obtained this group as a Galois group over C(t) using modular function fields 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 Furthermore in an unpublished preprint, he established that the corresponding Galois representation GC(t):=Gal(C(t)alg/C(t))→PSL2(Zp〚T〛) is universal.Here we will turn things around. We first provide a general framework for rigid deformations of (projective) representations of the absolute Galois group of a function field (in one variable) over a separably closed base. Under natural, rather general hypothesis, we will determine the corresponding universal deformation ring. If the residual representation is ‘geometrically rigid,’ which happens to be the case for many surjective representation to PSL2(Fp), p>2, which arise from Belyi triples, then certain universal deformations will be ‘geometrically rigid,’ too. This will give new proofs for most of the results of Rohrlich. Our method also applies to Thompson tuples.We then go on to give two further applications, which are based on the example computed by Rohrlich. Over Fq(t), where q is a power of a prime l, we construct infinite p-adic Galois extensions which have finite ramification and whose constant field is finite. Furthermore for p>5 and p≡1(mod4), we obtain a family of surjective Galois representations ρζ:Gal(Qalg/Q(ζ+ζ−1))↠SL2(Zp[ζ+ζ−1]), where the parameter ζ runs over all p-power roots of unity. Finally, we exhibit a general class of rigid universal deformation rings which are finite flat over Zp. In particular this shows that the above examples ρζ of Galois representations are not a singular event, but a general phenomenon