Let p be a rational prime. Let F be a totally real number field such that F is unramified over p and the residue degree of any prime ideal of F dividing p is $\leq 2$. In this paper, we show that the eigenvariety for $\mathrm {Res}_{F/\mathbb {Q}}(\mathit {GL}_{2})$, constructed by Andreatta, Iovita, and Pilloni, is proper at integral weights for $p\geq 3$. We also prove a weaker result for $p=2$.

Let S be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb {Z}_{S}$ -points on integral models of Hilbert modular varieties, extending a result of D. Helm and F. Voloch about modular curves. Let L be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre’s conjecture for mod $\ell $ representations of the absolute Galois group of L, we prove that the same holds also for the $\mathcal {O}_{L,S}$ -points.

We generalise results of Buzzard, Taylor and Kassaei on analytic continuation of p-adic overconvergent eigenforms over ℚ to the case of p-adic overconvergent Hilbert eigenforms over totally real fields F, under the assumption that p splits completely in F. This includes weight-one forms and has applications to generalisations of Buzzard and Taylor’s main theorem. Next, we follow an idea of Kassaei’s to generalise Coleman’s well-known result that ‘an overconvergent Up-eigenform of small slope is classical’ to the case of p-adic overconvergent Hilbert eigenforms of Iwahori level.

Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.