Spectral halo for Hilbert modular forms

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-p-slopes of points and the p-adic valuations of the p-parameters are bounded by explicit numbers, for all primes 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.

Automated summary

In this work, we prove that the eigenvariety associated to a definite quaternion algebra over a totally real field satisfies a specific property on the boundary annulus of the weight space. By applying Hansen's p-adic interpolation theorem, we extend our results to Hilbert modular eigenvarieties and show that the U-p slope of points approaches zero as they move towards the boundary on every irreducible component. This completes the proof of Coleman-Mazur's 'halo' conjecture, particularly in the case of eigencurves.