Deformations of pseudorepresentations (Q5942983)

If \(\rho: G\rightarrow {\text{GL}}(2,B)\) is a group homomorphism (\(B\) a commutative ring with identity), one can encode the semisimple type of \(\rho\) by memorizing the upper left and lower right entries of the matrices \(\rho(g),\;g\in G.\) This gives two maps \(A,D : G\rightarrow B\) which satisfy certain rules coming from matrix multiplication. A pseudorepresentation of \(G\) with values in \(B\) is a pair \(\pi\) of mappings \(A,D\) from \(G\) to \(B\) satisfying these rules. Pseudorepresentations can be composed with group homomorphisms on the one side and ring homomorphisms on the other side. NEWLINENEWLINENEWLINENow let \(G\) be the Galois group of the maximal extension of the rationals which is unramified outside some given set \(S\) of primes including \(\infty\), and specify a complex conjugation \(c\) in \(G\). Let \(k\) be a finite field of characteristic \(\neq 2.\) The author studies lifts of a pseudorepresentation \(\overline\pi\) of \(G\) with values in \(k\) and \(\overline A (c) = 1,\;\overline B(c) = -1\) to pseudorepresentations with values in some complete local noetherian ring \(B\) with residue field \(k\) and \(A(c) = 1,\;B(c) = -1\). There exist a universal deformation and a universal deformation ring \(R_{\overline\pi}\). This is the technical advance of pseudorepresentation in contrast to representations: Let \(\overline\rho:G\rightarrow {\text{GL}}(2,k)\) be a homomorphism inducing \(\overline\pi.\) Then \(\overline\rho\) has a versal deformation and a versal deformation ring \(R_{\overline\rho}.\) There is a natural homomorphism \(\phi:R_{\overline\pi} \rightarrow R_{\overline\rho}.\) If \(\overline\rho\) is absolutely irreducible, then \(R_{\overline\rho}\) is universal as well, and \(\phi\) is an isomorphism. NEWLINENEWLINENEWLINEThe main results in the final section treat the case where \(\overline\rho\) is not absolutely irreducible. In particular, \(R_{\overline\pi}\) is calculated for some classes of pseudorepresentations. These results are well prepared by detailed calculations for a specific example in section 3.