On locally analytic vectors of the completed cohomology of modular curves

Lue Pan

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arXiv · 2008.07099

On locally analytic vectors of the completed cohomology of modular curves

Lue Pan

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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.

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Lue Pan. 2020-08-17. On locally analytic vectors of the completed cohomology of modular curves. https://arxiv.org/abs/2008.07099

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