Lower bound on Néron-Tate height of Abelian varieties

TL;DR

For an abelian variety $A/K$ with symmetric ample line bundle and canonical height $\\hat{h}$, the paper proves a uniform Bogomolov-type lower bound for non-torsion points in an asymptotically positive extension $\\mathcal{K}/K$: $\\liminf_{P\in A(\\mathcal{K})}\\hat{h}(P) \\ge C(A,K) \\sum_{q= p^k}' \\psi_q \\frac{\\log q}{q^{2g}}$ (and a specialized form $\\liminf_{P\in A(\\mathcal{K})}\\hat{h}(P) \\ge C(A,K) \\psi_{p^f} \\frac{\\log p^f}{(p^f)^{2g}}$ for a fixed $p^f$). The approach combines an auxiliary section $F$ vanishing to high order at the origin (constructed via Siegel's lemma and differential-operator techniques), explicit height estimates, Liouville-type arguments, and Philippon's zero estimate to control small-height points; this yields Bogomolov property for totally $p$-adic points in the good-reduction case and implies finiteness of torsion in $A(\\mathcal{K})$. The work extends height-lower-bound phenomena beyond previously known bad-reduction settings and provides quantitative, extension-dependent lower bounds essential for understanding the distribution of low-height points in infinite extensions.

Abstract

Let $A$ be an abelian variety defined over a number field $K$ and $\hat{h}$ be the Néron-Tate height on $A(\overline{K})$ corresponding to a symmetric ample line bundle on $A$. Let $\mathcal{K}/K$ be an asymptotically positive infinite extension as defined in \cite{AB-SK} which includes infinite Galois extensions with finite local degree at a non-archimedean place. In this article, we prove that the Néron-Tate height of non-torsion points in $A(\mathcal{K})$ is bounded below by an absolute constant depending only on $A$, $K$, and $\mathcal{K}$. As a consequence, we obtain the Bogomolov property for totally $p$-adic points of an abelian variety $A/\mathbb{Q}$ in the case when $A$ has good reduction at $p$. This is the first instance where such a result has been obtained in the good reduction case; previously, it was known only in the bad reduction case via Gubler's tropical equidistribution theorem. Moreover, our result also implies the finiteness of torsion points in $A(\mathcal{K})$.

Theorems & Definitions (23)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4: S. Zhang Zhang
• Theorem 1.5: Baker-Silverman Baker-Silverman
• Theorem 1.6: Gubler Gubler
• Theorem 2.1
• Proposition 2.2
• Proposition 2.3
• proof