This content will become publicly available on February 17, 2023
Big images of twodimensional pseudorepresentations
Bellaïche has recently applied PinkLie theory to prove that, under mild conditions, the image of a continuous 2dimensional pseudorepresentation ρ of a proﬁnite group on a local pro p domain A contains a nontrivial congruence subgroup of SL2(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 selftwists of ρ, symmetries that constrain its image, and in terms of the adjoint trace ring of ρ, 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 bigimage results for GL2Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and padic Hida or Coleman families of elliptic and Hilbert modular forms.
 Award ID(s):
 1703834
 Publication Date:
 NSFPAR ID:
 10319140
 Journal Name:
 Mathematische Annalen
 ISSN:
 00255831
 Sponsoring Org:
 National Science Foundation
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