  We p-adically interpolate the relative de Rham cohomology of the universal
elliptic curve over strict neighbourhoods of the ordinary locus of modular
curves, together with the Hodge filtration and Gauss-Manin connection. Sections
of these sheaves provide the so called nearly overconvergent modular forms.
This extends previous work of Andreatta, Iovita and Pilloni where we p-adically
interpolate powers of the Hodge bundle and in that case the sections coincide
with Coleman overconvergent modular forms. We also show that, under suitable
assumptions, one can p-adically interpolate the Gauss-Manin connection. This is
used to define p-adic L-functions attached to a triple of p-adic finite slope
families of modular forms, generalizing previous constructions for Hida
families.
