We construct anticyclotomic p-adic L-functions of Hida families in a more controlled way using a multiplicity one result arising from arithmetic of Shimura curves. Using this construction, we can calculate the difference of Iwasawa invariants of p-adic L-functions of congruent modular forms in different weights. As an application, we can see the equivalence of the main conjectures of two congruent forms under certain conditions. This is joint work in progress with Francesc Castella and Matteo Longo.