In this work we prove various cases of the so-called “torsion congruences” between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the...