The valuative criterion for properness for eigenvarieties

Abstract: The question of whether the Coleman-Mazur eigencurve satisfies the valuative criterion for properness 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 short and explicit and uses no Galois theory. Instead we adapt an earlier method of Buzzard and Calegari based on elementary properties of overconvergent modular forms, for which we have to extend Pilloni's geometric construction of overconvergent forms of arbitrary weight farther into the supersingular locus. We will also discuss generalizations in progress.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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