A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety

Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna – the oft-termed BDP formula – we address questions about the non-vanishing of non-torsion points under $p$-adic logarithms of abelian varieties. We largely consider situations most applicable to ${\mathrm{G}L}_2$-type abelian varieties associated with Hilbert modular newforms and Heegner points. Not surprisingly, the main tool employed is the $p$-adic analytic subgroup theorem.