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Explicit universal deformations of Galois representations. (English) Zbl 0739.11021
Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values $$L$$-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 1-21 (1989).
[For the entire collection see Zbl 0721.00006.]
Let $$G_ \mathbb{Q}$$ be the Galois group of the algebraic closure of the field $$\mathbb{Q}$$ of rational numbers. Let (1) $$\bar\rho: G_ \mathbb{Q}\to GL_ 2(\mathbb{F}_ p)$$ be a continuous absolutely irreducible representation, and let $$S$$ be a finite set of primes which contains the primes of ramification for $$\bar\rho$$ and the prime number $$p$$. The authors study the universal deformation for $$(\bar\rho,S)$$. B. Mazur showed [in Galois groups over $$\mathbb{Q}$$, Publ., Math. Sci. Res. Inst. 16, 385-437 (1989; Zbl 0714.11076)] that there is a complete Noetherian local ring $$R$$ with residue field $$\mathbb{F}_ p$$, uniquely determined up to canonical isomorphism, and a lifting $$\rho: G_ \mathbb{Q}\to GL_ 2(R)$$ of $$\bar\rho$$ which is unramified outside $$S$$, and satisfies a universal property.
The object of the present paper is the study of the universal deformation space and the structure of its natural subspaces for the class of admissible $$S_ 3$$-representations. A representation (1) is called an admissible $$S_ 3$$-representation if it factors through an $$S_ 3$$- extension $$L/\mathbb{Q}$$ such that $$L$$ is the Galois closure of a cubic field $$K$$ which is not totally real and $$p$$ is the product of two prime ideals $$P_ 1$$, $$P_ 2$$ of degree 1. The authors distinguish generic and degenerate representations. In the generic case they determine the structure of the natural subspaces completely.
Reviewer: H.Koch (Berlin)

##### MSC:
 11F80 Galois representations 11F85 $$p$$-adic theory, local fields 11R32 Galois theory 14D15 Formal methods and deformations in algebraic geometry