2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/109124One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real number field.This is the major update to my deformation ring paper, which is already submitted to a journalNumber Theory11F55, 11F80, 11G18text