On quaternionic ordinary families of modular forms and $p$-adic $L$-functions

We use Serre–Tate expansions of modular forms to construct power series attached to quaternionic ordinary families of modular forms. We associate to these power series a big $p$-adic $L$-function interpolating the $p$-adic $L$-functions constructed by Burungale and Magrone at classical specializations. A crucial ingredient is the generalization of some results of Ohta to the quaternionic setting.