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

TL;DR

This paper investigates the non-vanishing of $p$-adic logarithms of rational points on abelian varieties, especially those related to Hilbert modular forms and Heegner points, using $p$-adic analytic subgroup techniques.

Contribution

It provides new non-vanishing results for $p$-adic logarithms of points on ${ m GL}_2$-type abelian varieties linked to modular forms and Heegner points.

Findings

Establishes conditions for non-vanishing of $p$-adic logarithms.

Applies $p$-adic analytic subgroup theorem to abelian varieties.

Focuses on abelian varieties associated with Hilbert modular forms.

Abstract

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 ${\mathrm GL}_2$-type abelian varieties associated with Hilbert modular newforms and Heegner points. Not surprisingly, the main tool employed is the $p$-adic analytic subgroup theorem.