Presentations of universal deformation rings

Let F be a finite field of characteristic ` > 0, F a number field, GF the absolute Galois group of F and let ρ̄ : GF → GLN (F) be an absolutely irreducible continuous representation. Suppose S is a finite set of places containing all places above ` and above ∞ and all those at which ρ̄ ramifies. Let O be a complete discrete valuation ring of characteristic zero with residue field F. In such a situation one may consider all deformations of ρ̄ to O-algebras which are unramified outside S and satisfy certain local deformation conditions at the places in S. This was first studied by Mazur, [12], and under rather general hypotheses, the existence of a universal deformation ring was proven. In [2] I studied, among other things, the number of generators needed for an ideal I in a presentations of such a universal deformation ring as a quotient of a power series ring over O by I. The present manuscript is an update of this part of [2]. The proofs have been simplified, the results slightly generalized. We also treat ` = 2, more general groups than GLN , and cases where not all relations are local. The results in [2] and hence also in the present manuscript are one of the (many) tools used in the recent attacks on Serre’s conjecture by C. Khare and others.