A Lehmer-type height lower bound for abelian surfaces over function fields

TL;DR

This paper establishes a lower bound for the canonical height of points on abelian surfaces over function fields, extending Lehmer's conjecture to this setting under mild assumptions.

Contribution

It proves a Lehmer-type height lower bound for points on abelian surfaces over function fields, a novel extension in the context of algebraic geometry and number theory.

Findings

Lower bound for the normalized Bernoulli-part of the canonical height.

Bound depends inversely on the square of the degree of the field extension.

Applicable to points with small positive height.

Abstract

Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(\bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ \hat{h}_A^{\mathbb{B}}(P) \ge C_1\bigl[K(P):K\bigr]^{-2} $$ for all points $P\in{A(\bar{K})}$ whose height satisfies $0<\hat{h}_A(P)\le{C_2}$.