In this talk, we consider Dirichlet forms related to stochastic quantisation for the \(exp(φ)_{2}\)-model on the torus. We show strong uniqueness of the corresponding Dirichlet operators by applying an idea of (singular) SPDEs. This talk is based on ongoing joint work with Hirotatsu Nagoji (Kyoto University).

Let $p$ be an odd prime. Let $K/\mathbf{Q}_p$ be a finite unramified extension. Let $\rho: G_K \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ be a continuous representation. We prove that $\rho$ has a crystalline lift of small irregular weight if and only if it has multiple crystalline lifts of certain specified regular weights. The inspiration for this result comes from recent work of Diamond and Sasaki on geometric Serre weight conjectures. We also discuss applications to partial weight one modularity.