Overconvergent Eichler–Shimura isomorphisms for quaternionic modular forms over ℚ

Abstract

In this work, we construct overconvergent Eichler–Shimura isomorphisms over Shimura curves over [Formula: see text]. More precisely, for a prime [Formula: see text] and a wide open disk [Formula: see text] in the weight space, we construct a Hecke–Galois-equivariant morphism from the space of families of overconvergent modular symbols over [Formula: see text] to the space of families of overconvergent modular forms over [Formula: see text]. In addition, for all but finitely many weights [Formula: see text], this morphism provides a description of the finite slope part of the space of overconvergent modular symbols of weight [Formula: see text] in terms of the finite slope part of the space of overconvergent modular forms of weight [Formula: see text]. Moreover, for classical weights these overconvergent isomorphisms are compatible with the classical Eichler–Shimura isomorphism.