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

💡 Research Summary

The paper investigates the non‑vanishing of p‑adic logarithms of rational points on abelian varieties, motivated by the Bertolini‑Darmon‑Prasanna (BDP) formula. The BDP formula relates a p‑adic L‑function attached to a weight‑2 Hilbert modular newform f to the square of the p‑adic Abel–Jacobi image of a generalized Heegner cycle; in the simplest case it reads

 Lₚ(f,K) = (1 – aₚ(f) + p⁻¹)²·(log_{ω_{B_f}}(P_f))²,

where K is an imaginary quadratic field satisfying the Heegner hypothesis for f, B_f is the GL₂‑type abelian variety attached to f, P_f ∈ B_f(K) is the Heegner point, and ω_{B_f} is the distinguished invariant differential.

If P_f is non‑torsion (equivalently L(f/K,s) has a simple zero at s=1 by Gross–Zagier), one naturally asks whether its p‑adic logarithm log_{ω_{B_f}}(P_f) is non‑zero. For elliptic curves the answer is trivial because the kernel of the p‑adic logarithm consists only of torsion points. For higher‑dimensional GL₂‑type abelian varieties, however, the question remained open.

The authors give an affirmative answer in full generality, using the p‑adic analytic subgroup theorem (a deep result of transcendence theory). Their main theorem (Theorem 1.1, restated as Theorem 2.3) says: let A/ℚ be an abelian variety equipped with an embedding of a number field F into End⁰_ℚ(A) such that dim A =