Can a family of finite-slope modular Hecke eigenforms lying over a punctured disc in weight space always be extended over the puncture? This was first asked by Coleman and Mazur in 1998 and settled by Diao and Liu in 2016 using deep, powerful Galois-theoretic machinery. We will discuss a new proof which is geometric and explicit and uses no Galois theory, and which generalizes in some cases to Hilbert modular forms. We adapt an earlier method of Buzzard and Calegari based on elementary properties of overconvergent modular forms, for which we have to extend the construction of overconvergent forms farther into the supersingular locus.
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