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 (in the affirmative) by Diao and Liu in 2014 using deep, powerful Galois-theoretic machinery. We will discuss a generalization of this result to Hilbert modular eigenvarieties for totally split primes. We do not use Diao-Liu's method. Instead we adapt an earlier method of Buzzard and Calegari based on elementary properties of overconvergent modular forms, building on recent work of Ren-Zhao for the boundary of weight space and Hattori for algebraic weights.
3/22/2021 - Hyuk Jun Kweon (MIT) - Bounds on the Torsion Subgroups of Néron-Severi Group Schemes