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

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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 that 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.