Big images of two-dimensional pseudorepresentations

Abstract

Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation $$\rho $$ of a profinite group on a local pro-p domain A contains a nontrivial congruence subgroup of $${{\,\mathrm{{SL}}\,}}_2(B)$$ for a certain subring B of A. We enlarge Bellaïche’s ring and give this new B a conceptual interpretation both in terms of conjugate self-twists of $$\rho $$ , symmetries that constrain its image, and in terms of the adjoint trace ring of $$\rho $$ , which we show is both more natural and the optimal ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for $${{\,\mathrm{GL}\,}}_2$$ Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida or Coleman families of elliptic and Hilbert modular forms.