Several variables $p$-adic $L$-functions for Hida families of Hilbert modular forms

Abstract

After formulating Conjecture A for $p$-adic $L$-functions defined over ordinary Hilbert modular Hida deformations on a totally real field $F$ of degree $d$, we construct two $p$-adic $L$-functions of $d+1$-variable depending on the parity of weight as a partial result on Conjecture A. We will also state Conjecture B which is a corollary of Conjecture A but is important by itself. Main issues of the construction are the study of Hida theory of Hilbert modular forms by using Hilbert modular varieties (without using Shimura curves), the study of higher dimensional modular symbols on Hilbert modular varieties and delicate treatments on archimedean and $p$-adic periods.