Finiteness for crystalline representations of the absolute Galois group of a totally real field

Let K be a totally real field and G_K:=Gal(K‾/K) its absolute Galois group, where K‾ is a fixed algebraic closure of K. Let ℓ be a prime and E a finite extension of Q_ℓ. Let S be a finite set of finite places of K not dividing ℓ. Assume that K, S, Hodge-Tate type h and a positive integer n are fixed. In this paper, we prove that if ℓ is sufficiently large, then, for any fixed E, there are only finitely many isomorphism classes of crystalline representations r:G_K→GL_n(E) unramified outside S∪{v:v|ℓ}, with fixed Hodge-Tate type h, such that r|_GK^′ ≃⊕r_i ^′ for some finite totally real field extension K^′ of K unramified at all places of K over ℓ, where each representation r_i ^′ over E is an 1-dimensional representation of G_K^′ or a totally odd irreducible 2-dimensional representation of G_K^′ with distinct Hodge-Tate numbers.