We generalise results of Buzzard, Taylor and Kassaei on analytic continuation of padic overconvergent eigenforms over Q to the case of p-adic overconvergent Hilbert eigenforms over totally real fields F , under the assumption that p splits completely in F . This includes weight-one forms and has applications to generalisations of Buzzard and Taylor's main theorem. Next, we follow an idea of Kassaei's to generalise Coleman's well-known result that 'an overconvergent U p -eigenform of small slope is classical' to the case of p-adic overconvergent Hilbert eigenforms of Iwahori level.
We extend the modularity lifting result of P. Kassaei ('Modularity lifting in parallel weight one', J. Amer. Math. Soc. 26 (1) (2013), 199-225) to allow Galois representations with some ramification at p. We also prove modularity mod 5 of certain Galois representations. We use these results to prove new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in P. Kassaei ('Modularity lifting in parallel weight one',
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