On a properness of the Hilbert eigenvariety at integral weights: the case of quadratic residue fields

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 1 or 2. In this paper, we show that the eigenvariety for Res_{F/Q}(GL_2), constructed by Andreatta-Iovita-Pilloni, is proper at integral weights for p>=3. We also prove a weaker result for p=2.