In this work, we construct overconvergent Eichler–Shimura isomorphisms over Shimura curves over QQ. More precisely, for a prime p>3p>3 and a wide open disk UU in the weight space, we construct a Hecke–Galois-equivariant morphism from the space of families of overconvergent modular symbols over UU to the space of families of overconvergent modular forms over UU. In addition, for all but finitely many weights ¿¿U¿¿U, this morphism provides a description of the finite slope part of the space of overconvergent modular symbols of weight ¿¿ in terms of the finite slope part of the space of overconvergent modular forms of weight ¿+2¿+2. Moreover, for classical weights these overconvergent isomorphisms are compatible with the classical Eichler–Shimura isomorphism.
In this paper we prove a version of Deligne's conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.