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

TL;DR

This paper explores how geometric translations of $(, )$-modules over the Robba ring relate to changes in weights, aiming to understand their impact on locally analytic representations within the $p$-adic Langlands framework for $ ext{GL}_2(Q_p)$.

Contribution

It demonstrates that in the $ ext{GL}_2(Q_p)$ case, change of weights maps can be realized as geometric translations of $(, )$-modules, linking geometric and representation-theoretic perspectives.

Findings

Change of weights maps can be geometrically realized for $ ext{GL}_2(Q_p)$.

Provides a geometric interpretation of weight translations in the $(, )$-module stack.

Enhances understanding of the categorical $p$-adic Langlands correspondence.

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.