Geometric translations of $(\varphi,\Gamma)$-modules for $\mathrm{GL}_2(\mathbb{Q}_p)$

Abstract

We study ``change of weights'' maps between loci of the stack of $(\varphi,\Gamma)$-modules over the Robba ring with integral Hodge-Tate-Sen weights. We show that in the $\mathrm{GL}_2(\mathbb{Q}_p)$ case these maps can realize translations of $(\varphi,\Gamma)$-modules geometrically. The motivation is to investigate translations of locally analytic representations under the categorical $p$-adic Langlands correspondence.