Modularity of trianguline Galois representations

TL;DR

This paper proves a modularity lifting theorem for trianguline Galois representations using pseudorigid spaces and Taylor--Wiles patching, advancing the understanding of Galois representations with characteristic p coefficients.

Contribution

It introduces a novel approach employing pseudorigid spaces to construct integral models of trianguline varieties and applies Taylor--Wiles patching to establish modularity lifting results.

Findings

Established a modularity lifting theorem for trianguline Galois representations.

Constructed a patched quaternionic eigenvariety.

Extended modularity results to representations with characteristic p coefficients.

Abstract

We use the theory of trianguline $(\varphi,\Gamma)$-modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at $p$, including those with characteristic $p$ coefficients. The use of pseudorigid spaces lets us construct integral models of the trianguline varieties of [BHS17b], [Che13] after bounding the slope, and we carry out a Taylor--Wiles patching argument for families of overconvergent modular forms. This permits us to construct a patched quaternionic eigenvariety and deduce our modularity results.