Spectral halo for Hilbert modular forms (Q2134205)

The theory of $p$-adic analytic families of modular forms is a result of study of congruences of modular forms. The first example of such a family was given by \textit{J.-P. Serre} [Lect. Notes Math. 350, 191--268 (1973; Zbl 0277.12014)]. After a long time, Hida started the first attempt towards the construction of $p$-adic families of cuspforms [\textit{H. Hida}, Invent. Math. 85, 545--613 (1986; Zbl 0612.10021); Ann. Sci. Éc. Norm. Supér. (4) 19, No. 2, 231--273 (1986; Zbl 0607.10022)]. Hida's theory led Mazur to develop his general theory of deformations of Galois representations, which turns out to be a crucial ingredient of Wiles' proof of Fermat's last theorem [\textit{B. Mazur}, Publ., Math. Sci. Res. Inst. 16, 385--437 (1989; Zbl 0714.11076)]. This theory can be found in the celebrated work of Coleman and Mazur in which they constructed eigencurves [\textit{R. Coleman} and \textit{B. Mazur}, Lond. Math. Soc. Lect. Note Ser. 254, 1--113 (1998; Zbl 0932.11030)]. In the geometric description of the of the eigencurves, Coleman and Mazur raised a question that is turned out in the following conjecture: ``It is expected that over the boundary of the weight space, the eigencurve is a disjoint union of infinitely many connected components that are finite flat over the weight space [\textit{K. S. Kedlaya}, \(p\)-adic differential equations. Cambridge: Cambridge University Press (2010; Zbl 1213.12009)].'' The authors of the paper under review generalize the results in [\textit{R. Liu} et al., Duke Math. J. 166, No. 9, 1739--1787 (2017; Zbl 1423.11089)] to the eigenvarieties associated to $p$-adic overconvergent automorphic forms for a definite quaternion algebra over a totally real field $F$ in which $p$ splits completely. Combining with the $p$-adic family versions of base change and Jacquet-Langlands correspondence, this result allows them to determine the boundary behavior for the entire Coleman-Mazur eigencurves. The paper is a long well written paper and is an excellent source for interested researchers in the field.