In the paper at hand the author studies slopes of \(p\)-adic Hilbert modular forms. The slope of an eigenform is the eigenvalue corresponding to the \(U_p\)-operator, and for modular forms for \(\mathrm{GL}_2/\mathbb{Q}\) the slopes of a family of modular forms, parametrized by the weight, have a very rich structure. For example, in some particular cases and near the boundary of the weight space, the slopes are in arithmetic progression. Similar results are known for modular forms associated with other groups, for example quaternionic modular forms. These properties have important consequences about the Galois representation associated to modular forms and so it is natural to try to generalize them to other groups.
In this paper, the author studies the situation for Hilbert modular forms. The weight space is more complicated and much less is known in general. In the special case of Hilbert modular forms for \(\mathbb{Q}(\sqrt 5)\) and for \(p=2\), the author shows that the slopes of \(U_p\) are completely determined by slopes of classical Hilbert modular forms (for a specific tame level). This gives a better explanation of why the slopes are in arithmetic progression in the elliptic case.
The techniques used to prove the main result are quite ad hoc, but are very explicit and they allow the author to formulate a conjecture about the structure of the eigenvariety similar to what is already known in the elliptic case.