Modularity of trianguline Galois representations

Abstract

We use the theory of trianguline (ϕ, Γ)-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.