On locally analytic vectors of the completed cohomology of modular curves II

TL;DR

This paper extends the understanding of locally analytic vectors in the completed cohomology of modular curves by constructing differential operators and providing new geometric insights into associated Galois representations.

Contribution

It introduces differential operators on modular curves with infinite p-level and offers a geometric description of related locally analytic representations.

Findings

Reproved Emerton's classicality result for certain Galois representations.

Constructed differential operators in both holomorphic and anti-holomorphic directions.

Provided a geometric interpretation of locally analytic representations attached to Galois representations.

Abstract

This is a continuation of our previous work on the locally analytic vectors of the completed cohomology of modular curves. We construct differential operators on modular curves with infinite level at p in both "holomorphic" and "anti-holomorphic" directions. As applications, we reprove a classicality result of Emerton which says that every absolutely irreducible two dimensional Galois representation which is regular de Rham at p and appears in the completed cohomology of modular curves comes from an eigenform. Moreover we give a geometric description of the locally analytic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ attached to such a Galois representation in the completed cohomology.