A modular proof of the properness of the Coleman-Mazur eigencurve

TL;DR

This paper provides a new, simpler proof of the properness of the Coleman-Mazur eigencurve, avoiding deep Galois theory by using elementary properties of overconvergent modular forms and extending geometric constructions.

Contribution

It introduces a modular, explicit proof of properness that extends Pilloni's construction into the supersingular locus and demonstrates injectivity of the $U_p$ operator.

Findings

The eigencurve satisfies the valuative criterion for properness.

The $U_p$ operator is injective on certain spaces of overconvergent forms.

The proof avoids Galois theory and relies on elementary modular form properties.

Abstract

We give a new proof of the properness of the Coleman-Mazur eigencurve. The question of whether the 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 Hodge- and Galois- theoretic machinery. Our proof 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. To facilitate this, we extend Pilloni's geometric construction of overconvergent forms of arbitrary weight farther into the supersingular locus. Along the way, we show that the Hecke operator $U_p$ is injective on spaces of forms of large overconvergence radius of any analytic weight.