15 June 2017 The eigencurve over the boundary of weight space
Ruochuan Liu, Daqing Wan, Liang Xiao
Duke Math. J. 166(9): 1739-1787 (15 June 2017). DOI: 10.1215/00127094-0000012X


We prove that the eigencurve associated to a definite quaternion algebra over Q satisfies the following properties, as conjectured by Coleman and Mazur as well as Buzzard and 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 Up-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 forms a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves toward the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.


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Ruochuan Liu. Daqing Wan. Liang Xiao. "The eigencurve over the boundary of weight space." Duke Math. J. 166 (9) 1739 - 1787, 15 June 2017. https://doi.org/10.1215/00127094-0000012X


Received: 3 August 2015; Revised: 28 October 2016; Published: 15 June 2017
First available in Project Euclid: 4 April 2017

zbMATH: 06745538
MathSciNet: MR3662443
Digital Object Identifier: 10.1215/00127094-0000012X

Primary: 11F33
Secondary: 11F85 , 11S05

Keywords: completed cohomology , eigencurves , Newton polygon , overconvergent modular forms , slope of $U_{p}$-operators , weight space

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 9 • 15 June 2017
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