The eigencurve over the boundary of weight space
Open Access Publications from the University of California

## The eigencurve over the boundary of weight space

• Author(s): Liu, R
• Wan, D
• Xiao, L
• et al.

## Published Web Location

https://doi.org/10.1215/00127094-0000012X
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.

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