  In this work we prove 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 special cases of $n=1$ and $n=2$. In both
of these cases we also explain their implications for some particular
"motives", as for example elliptic curves with complex multiplication.
