p-adic interpolation of Gauss--Manin connections on nearly overconvergent modular forms and p-adic L-functions

In this paper, we give a new geometric definition of nearly overconvergent modular forms and
p-adically interpolate the Gauss-Manin connection on this space. This can be seen as an “overconvergent”
version of the unipotent circle action on the space of p-adic modular forms, as constructed by Gouvˆea and
Howe. This improves on results of Andreatta–Iovita and has applications to the construction of Rankin–
Selberg and triple product p-adic L-functions.