If the Albanese kernel of X×X is torsion, the generalised p-adic height pairing on rational points of the Jacobian admits an algorithmic computation, via relevance of the Beilinson-Bloch conjectures.
The polylog quotient and the Gon- charov quotient in computational Chabauty–Kim theory I
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Generalised height pairings and the Albanese kernel
If the Albanese kernel of X×X is torsion, the generalised p-adic height pairing on rational points of the Jacobian admits an algorithmic computation, via relevance of the Beilinson-Bloch conjectures.