Spectral halo for Hilbert modular forms

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.