Abstract
Let F be a totally real field and p be an odd prime which splits completely in F. We prove that the eigenvariety associated to a definite quaternion algebra over F satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the \(U_\mathfrak {p}\)-slopes of points and the p-adic valuations of the \(\mathfrak {p}\)-parameters are bounded by explicit numbers, for all primes \(\mathfrak {p}\) of F over p. Applying Hansen’s p-adic interpolation theorem, we are able to transfer our results to Hilbert modular eigenvarieties. In particular, we prove that on every irreducible component of Hilbert modular eigenvarieties, as a point moves towards the boundary, its \(U_p\) slope goes to zero. In the case of eigencurves, this completes the proof of Coleman–Mazur’s ‘halo’ conjecture.
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Acknowledgements
We thank Liang Xiao for sharing his ideas and answering many questions on this topic. We would also like to thank Yiwen Ding, Yongquan Hu and Daqing Wan for helpful comments and conversations. Finally we would like to thank the anonymous referees for their impressively helpful report that significantly improved the exposition of this paper.
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Communicated by Wei Zhang.
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Ren, R., Zhao, B. Spectral halo for Hilbert modular forms. Math. Ann. 382, 821–899 (2022). https://doi.org/10.1007/s00208-021-02184-9
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DOI: https://doi.org/10.1007/s00208-021-02184-9