The Sen-operator conjecture for rational Hodge–Tate crystals

Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field. Let ${\mathbb M}$ be a rational Hodge–Tate crystal in

$${\mathrm{Vect}}(({\mathcal O}_K)_{{\mathlarger{\mathbbl{\Delta}}}},\overline {\mathcal O}_{{\mathlarger{\mathbbl{\Delta}}}}[\frac{1}{p}])$$

with associated pair $(M,\phi_M)$, and let $V({\mathbb M})$ be the corresponding semi-linear ${\mathbb C}_p$-representation of $G_K$. The Sen-operator conjecture. The Sen operator of $V({\mathbb M})$ is

$$\Theta_{V({\mathbb M})}=\frac{-\phi_M}{E'(\pi)}.$$

The conjecture identifies the Sen operator of the representation attached to a rational Hodge–Tate crystal directly from its associated linear-algebraic data. It was proved by Hui Gao using Sen theory via locally analytic vectors.