# Big images of two-dimensional pseudorepresentations

## TL;DR

This paper extends Bella"iche's work on 2-dimensional pseudorepresentations, providing a conceptual interpretation of the image of such representations and applying the results to various arithmetic contexts involving modular forms.

## Contribution

It enlarges Bella"iche's ring, interprets it via conjugate self-twists, and connects it to the adjoint trace ring, offering a more natural framework for big-image results.

## Key findings

- The new ring $B$ is optimal among congruence subgroups in the image.
- The results recover and extend big-image theorems for Galois representations.
- The approach applies to elliptic, Hilbert, and Bianchi modular forms, and p-adic families.

## Abstract

Bella\"iche 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 ${\rm SL}_2(B)$ for a certain subring $B$ of $A$. We enlarge Bella\"iche's ring and give this new $B$ a conceptual interpretation in terms of conjugate self-twists of $\rho$, symmetries that naturally constrain its image. As a corollary, this new $B$ is optimal among congruence subgroups contained in the image. We also interpret the new $B$ vis-a-vis the adjoint trace ring of $\rho$, which we show is a more natural 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 ${\rm GL}_2$ Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and $p$-adic Hida or Coleman families of elliptic and Hilbert modular forms.