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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.03039 |
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Table of Contents:
- Let $v$ be a finite place of a number field $K$ and write $K^{nr,v}$ for the maximal field extension of $K$ in which $v$ is unramified. The purpose of this paper is split up into two parts. The first one generalizes a theorem of Pottmeyer: If $E$ is an elliptic curve defined over $K$ with split multiplicative reduction at $v$, then the Néron-Tate height of a non-torsion point $P\in E(\bar{K})$ is bounded from below by $C / e_v(P)^{2 e_v(P)+1}$, where $C>0$ is an absolute constant and $e_v(P)$ is the maximum of all ramification indices $e_w(K(P) \vert K)$ with $w\vert v$. Among other things, we refine this result by showing that given a simple abelian variety $A$ defined over $K$ that is degenerate at $v$, the Néron-Tate height of a non-torsion point $P\in A(\bar{K})$ is at least $C / \mathrm{lcm}_{w\vert v} \{e_w(K(P)\vert K)\}^2$, where $C>0$ is an absolute constant. We then give applications towards Lehmer's conjecture. Next, we provide the first examples of polynomials $ϕ\in K[X]$ of degree at least $2$ so that the canonical height $\hat{h}_ϕ$ of any point in $\bbP^1(K^{nr,v})$ is either $0$ or bounded from below by an absolute positive constant.