A canonical generator for congruence ideals of Hida families

We construct adjoint p -adic L -functions generating the congruence ideal attached to Hida families. These functions interpolate the Petersson norm of any classical ordinary newform, normalized by a product of Shimura’s canonical periods. We show that, after adjusting by suitable Euler factors, they are interpolated by a regular element of Hida’s universal ordinary Hecke algebra. We also establish a link between these p -adic L -functions and the characteristic series of primitive adjoint Selmer groups.