  Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an
odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under
not-too-stringent hypotheses we give an interpretation of the $p$-component of
the relevant case of the equivariant Tamagawa number conjecture in terms of
integral congruence relations involving the evaluation on appropriate points of
$A$ of the ${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then
discuss the numerical computation of this pairing, and in particular obtain the
first numerical verifications of this conjecture in situations in which the
$p$-completion of the Mordell-Weil group of $A$ over $F$ is not a projective
Galois module.
