In 1978, Katz gave a construction of the $p$-adic $L$-function of a CM field by using a $p$-adic analog of the Maass--Shimura operators acting on $p$-adic Hilbert modular forms. However, this $p$-adic Maass--Shimura operator is only defined over the ordinary locus, which restricted Katz's choice of $p$ to one that splits in the CM field. In 2021, Andreatta and Iovita extended Katz's construction to all $p$ for quadratic imaginary fields using overconvergent differential operators constructed by Harron--Xiao and Urban, which act on elliptic modular forms. Here we give a construction of such overconvergent differential operators which act on Hilbert modular forms.

[Pre-talk at 1:20PM]