Lifting Galois representations via Kummer flags

Abstract

Let $\Gamma$ be either i) the absolute Galois group of a local field $F$, or ii) the topological fundamental group of a closed connected orientable surface of genus $g$. In case i), assume that $\mu_{p^2} \subset F$. We give an elementary and unified proof that every representation $\rho_1: \Gamma \to \mathbf{GL}_d(\mathbb{F}_p)$ lifts to a representation $\rho_2: \Gamma \to \mathbf{GL}_d(\mathbb{Z}/p^2)$. [In case i), it is understood these are continuous.] The actual statement is much stronger: for all $r \geq 1$, under "suitable" assumptions, triangular representations $\rho_r: \Gamma \to \mathbf{B}_d(\mathbb{Z}/p^r)$ lift to $\rho_{r+1}: \Gamma \to \mathbf{B}_d(\mathbb{Z}/p^{r+1})$, in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of $\textit{Kummer flag}$. An essential aspect of this work is to identify the common properties of groups i) and ii) that suffice to ensure the existence of such lifts.