  We develop a framework to investigate conjectures on congruences between the
algebraic part of special values of $L$-functions of congruent motives. We show
that algebraic local Euler factors satisfy precise interpolation properties in
$p$-adic families of motives and that algebraic $p$-adic $L$-functions exist in
quite large generality for $p$-adic families of automorphic motives. We
formulate two conjectures refining (and correcting) the currently existing
formulation of the Equivariant Tamagawa Number Conjecture with coefficients in
Hecke algebras and pointing out the links between conjectures on special values
and completed cohomology.
