Slopes of modular forms and geometry of eigencurves

TL;DR

This paper proves a local analogue of the ghost conjecture under generic conditions, leading to several significant results about slopes of modular forms, eigencurves, and deformation spaces.

Contribution

It establishes the local ghost conjecture under stronger genericity, confirming multiple conjectures related to slopes and eigencurve geometry.

Findings

Proof of local ghost conjecture under generic conditions

Verification of folklore conjecture on crystalline slopes

Finiteness of irreducible components of eigencurves

Abstract

Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture.