Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III

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