##
**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 \(GL_{2,{\mathbb{Q}}})\).

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 |