A 0,5 (half) overconvergent Eichler-Shimura isomorphism

In this article we construct a Galois and Hecke equivariant morphism connecting the first cohomology group on Faltings’ site of a formal strict neighborhood of the ordinary locus in a formal modular curve of level prime to p, with coefficients in the analytic distributions of a certain analytic weight k on the p-adic Tate module of the universal elliptic curve to the overconvergent modular forms of weight k+ 2. We prove that this morphism is an isomorphism on the finite slope parts.