A canonical generator for congruence ideals of Hida families

We construct canonical adjoint $p$-adic $L$-functions generating the congruence ideal attached to Hida families using Ohta’s pairing. We show that these $p$-adic $L$-functions, suitably modified by certain Euler factors, are interpolated by a regular element of Hida’s universal ordinary Hecke algebra. We also relate them to characteristic series of primitive adjoint Selmer groups.