Abstract

In this work we construct overconvergent Eichler-Shimura isomorphisms on Shimura curves
over a totally real field F. More precisely, for a prime p > 2 and a wide open disk U in the
weight space, we construct a Hecke-Galois-equivariant morphism from the space of families
of overconvergent modular symbols over U to the space of families of overconvergent modular
forms over U. In addition, for all but finitely many weights λ ∈ U, this morphism provides a
description of the finite slope part of the space of overconvergent modular symbols of weight
λ in terms of the finite slope part of the space of overconvergent modular forms of weight
λ + 2. Moreover, for classical weights these overconvergent isomorphisms are compatible
with the classical Eichler-Shimura isomorphisms.