Overconvergent Eichler-Shimura isomorphisms

Given a prime p>2, an integer h≥0, and a wide open disk U in the weight space W of GL2, we construct a Hecke–Galois-equivariant morphism Ψ(h)U from the space of analytic families of overconvergent modular symbols over U with bounded slope ≤h, to the corresponding space of analytic families of overconvergent modular forms, all with Cp-coefficients. We show that there is a finite subset Z of U for which this morphism induces a p-adic analytic family of isomorphisms relating overconvergent modular symbols of weight k and slope ≤h to overconvergent modular forms of weight k+2 and slope ≤h.