On locally analytic vectors of the completed cohomology of modular curves

TL;DR

This paper investigates the structure of locally analytic vectors in the completed cohomology of modular curves, providing new insights into eigenvectors, classicality of overconvergent eigenforms, and Galois representations.

Contribution

It characterizes locally analytic vectors and eigenvectors in completed cohomology, proves classicality for weight one eigenforms, and offers a new proof of the Fontaine-Mazur conjecture in certain cases.

Findings

Eigenvectors of a rational Borel subalgebra are determined.

Classicality of overconvergent eigenforms of weight one is established.

Galois representations associated to overconvergent eigenforms have specific Hodge-Tate-Sen weights.

Abstract

We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight $k$, we show its corresponding Galois representation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.