Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms

Let $L$ be a totally real field, and $p$ be a rational prime that is unramified in $L$. We construct overconvergent families of classes of relative de Rham cohomology of the universal abelian scheme over Hilbert modular varieties associated to $L$. We show that these classes come equipped with Gauss-Manin connection. We prove convergence for $p$-adic iteration of this connection, improving upon a technique due to Andreatta-Iovita. We use this to construct a $p$-adic twisted triple product $L$-function associated to finite slope families of Hilbert modular forms, extending work of Blanco-Chacon-Fornea for Hida families.