Automorphic symbols, p-adic L-functions and ordinary cohomology of Hilbert modular varieties

We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general $p$-adic $L$-functions to nearly ordinary Hilbert automorphic forms. Then we establish an exact control theorem for the $p$-adically completed cohomology of a Hilbert modular variety localized at a suitable nearly ordinary maximal ideal of the Hecke algebra. We also show its freeness over the corresponding Hecke algebra which turns out to be a universal deformation ring. In the last part of the paper we combine the above results to construct $p$-adic $L$-functions for Hida families of Hilbert automorphic forms in universal deformation rings of Galois representations.