Images of two-dimensional pseudorepresentations

There is a general philosophy that the image of a Galois representation should be as large as possible, subject to its symmetries. This can be seen in Serre's open image theorem for non-CM elliptic curves, Ribet and Momose's work on Galois representations attached to modular forms, and recent work of the speaker and Conti-Iovita-Tilouine on Galois representations attached to p-adic families of modular forms. Recently, Bellaïche developed a way to measure the image of an arbitrary pseudorepresentations taking values in a local ring A. Under the assumptions that A is a domain and the residual representation is not too degenerate, we explain how the symmetries of such a pseudorepresentation are reflected in its image. This is joint work with Andrea Conti and Anna Medvedovsky.