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Slopes in eigenvarieties for definite unitary groups
Published online by Cambridge University Press: 22 November 2023
Abstract
We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank
$n$, the Newton polygon of the characteristic power series of the
$U_p$ Hecke operator has exact growth rate
$x^{1+2/{n(n-1)}}$, times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.
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- Research Article
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- Copyright
- © 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
Footnotes
This work was partially done with the support of a National Defense Science and Engineering Graduate Fellowship, and writing was completed with the support of a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship.
References
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