We study the $p$-adic variation of triangulations over $p$-adic families of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. We also study the ramification of weight parameters over natural $p$-adic families.

In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of kth invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.

For V a two-dimensional p-adic representation of Gℚp, we denote by B(V ) the admissible unitary representation of GL2(ℚp) attached to V under the p-adic local Langlands correspondence of GL2(ℚp) initiated by Breuil. In this paper, building on the works of Berger–Breuil and Colmez, we determine the locally analytic vectors B(V )an of B (V ) when V is irreducible, crystabelian and Frobenius semisimple with distinct Hodge–Tate weights; this proves a conjecture of Breuil. Using this result, we verify Emerton’s conjecture that dim Ref η⊗ψ (V )=dim Exp η∣⋅∣⊗xψ (B (V )an ⊗(x∣⋅∣∘det )) for those V which are irreducible, crystabelian and Frobenius semisimple.

We set up the basic theory of $\mathcal P$-adic modular forms over certain unitary PEL Shimura curves M′K′. For any PEL abelian scheme classified by M′K′, which is not ‘too supersingular’, we construct a canonical subgroup which is essentially a lifting of the kernel of Frobenius from characteristic p. Using this construction we define the U and Frob operators in this context. Following Coleman, we study the spectral theory of the action of U on families of overconvergent $\mathcal P$-adic modular forms and prove that the dimension of overconvergent eigenforms of U of a given slope is a locally constant function of the weight.