| Résume | To prove modularity of two-diemsnsional icosahedral Artin representations, Buzzard and Taylor observed that one could find a weight one form in p-adic families of modular forms, in particular, the rigid geometric framework which Coleman had developed in the 90s. One of the very technical observations they made was that an overconvegent $U_p$ eigeform extends across to the non-ordinary locus of a modular curve and thereby one can glue two overconvergent (companion) forms. Inspired by this, Kassaei subsequently reproved Coleman's result that, if its slope is small, only one overconvergent eigenform is needed to extend to the entire modular curve. I plan to present that, in fact, their techniques are applicable to the case of Hilbert modular forms over a totally real F, with the assumption that p splits completely in F. If there is time, I shall briefly mention what happens when p donesn't split. |
