The eigencurve over the boundary of weight space

Abstract

We prove that the eigencurve associated to a definite quaternion algebra over $\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the $U_p$-slopes of points on each fixed connected component are proportional to the $p$-adic valuations of the parameter on weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves towards the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.

36 pages. Final version, to appear in Duke Math Journal

References in corpus (3)

• Slopes for higher rank Artin-Schreier-Witt Towers
• Arithmetic properties of Fredholm series for p-adic modular forms
• The Eigencurve is Proper

Cited by in corpus (13)

• Extended eigenvarieties for overconvergent cohomology
• Arithmetic properties of Fredholm series for p-adic modular forms
• Slopes of modular forms and the ghost conjecture
• Slopes of modular forms and the ghost conjecture, II
• Slopes of modular forms and the ghost conjecture (unabridged version)
• Equidimensional adic eigenvarieties for groups with discrete series
• The Jacquet-Langlands correspondence for overconvergent Hilbert modular forms
• Upper bounds for constant slope $p$-adic families of modular forms
• Slopes of overconvergent Hilbert modular forms
• Newton Polygons of Hecke Operators
• Overconvergent quaternionic forms and anticyclotomic p-adic L-functions
• $2$-adic slopes of Hilbert modular forms over $\mathbb{Q}(\sqrt{5})$
• An explicit computation of the Hecke operator and the ghost conjecture