Lifting Galois representations via Kummer flags

Abstract

Let $\hat{I}“$ 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 $\hat{I}{¼}_{p^2}\subset F$. We give an elementary and unified proof that every representation $\ddot{I}{}_1:\hat{I}“\to {\mathbf{GL}}_d({\mathbb{F}}_p)$ lifts to a representation $\ddot{I}{}_2:\hat{I}“\to {\mathbf{GL}}_d(\mathbb{Z}/p^2)$. [In case i), it is understood these are continuous.] The actual statement is much stronger: for all $r\ge 1$, under "suitable" assumptions, triangular representations $\ddot{I}{}_r:\hat{I}“\to {\mathbf{B}}_d(\mathbb{Z}/p^r)$ lift to $\ddot{I}{}_{r+1}:\hat{I}“\to {\mathbf{B}}_d(\mathbb{Z}/p^{r+1})$, in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of $\text{Kummer flag}$. 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.

29 pages. We fixed a problem in the last case of the proof of Theorem 7.21