Geodesic
Modularity of trianguline Galois representations
Forum of Mathematics, Sigma, Tome 12 (2024)

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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 [BHS17], [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. 
@article{10_1017_fms_2023_116,
     author = {Rebecca Bellovin},
     title = {Modularity of trianguline {Galois} representations},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {12},
     year = {2024},
     doi = {10.1017/fms.2023.116},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.116/}
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Rebecca Bellovin. Modularity of trianguline Galois representations. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2023.116

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