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.
Hyperelliptic curves mapping to abelian varieties and applications to beilinson’s conjecture for zero-cycles.arXiv preprint arXiv:2309.06361,
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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.