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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).

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[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)

11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11R32 Galois theory
14D15 Formal methods and deformations in algebraic geometry