Deforming Galois representations.(English)Zbl 0714.11076

Galois groups over $${\mathbb{Q}}$$, Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 385-437 (1989).
[For the entire collection see Zbl 0684.00005.]
Let S be a finite set of rational primes and $$G_{{\mathbb{Q}},S}$$ the Galois group over $${\mathbb{Q}}$$ of a maximal algebraic field extension that is unramified outside S. Consider a continuous representation $${\bar \rho}$$: $$G_{{\mathbb{Q}},S}\to GL_ N(k)$$ over a finite field k of characteristic p. Given a complete Noetherian local ring A with residue field k, a deformation of $${\bar \rho}$$ is, by definition, a continuous representation $$G_{{\mathbb{Q}},S}\to GL_ N(A)$$ which reduces to $${\bar \rho}$$ modulo the maximal ideal of A. Two deformations are considered equal if these representations are conjugate under $$\ker (GL_ n(A)\to GL_ n(k)).$$
The first half of the article develops an abstract theory of such deformations. Assume that $${\bar \rho}$$ is absolutely irreducible. The author proves the existence of a universal deformation $$\rho$$ : $$G_{{\mathbb{Q}},S}\to GL_ N(R)$$. That is, every other deformation comes from combining $$\rho$$ with a unique ring homomorphism $$R\to A$$. Basic properties of R are investigated: functoriality, tangent space, Krull dimension.
The set of all deformations of $${\bar \rho}$$ can be studied in the language of algebraic geometry on the universal deformation space $$X=Spec R$$. In particular, deformations with certain additional properties (e.g. on the structure of the image of $$G_{{\mathbb{Q}},S}$$ or of an inertia subgroup) correspond to strata in X. Geometric properties of such strata are analyzed in detail in the second half of the article, for certain special $${\bar \rho}$$ of dimension $$N=2$$ whose image is a dihedral group of order prime to p. In these cases R is isomorphic to a power series ring in three variables over $${\mathbb{Z}}_ p$$. An important role is played by the Galois representations associated to automorphic forms (on $$GL_{2,{\mathbb{Q}}})$$.
Reviewer: R.Pink

MSC:

 11R32 Galois theory 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S20 Galois theory 14D15 Formal methods and deformations in algebraic geometry

Zbl 0684.00005