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

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 $${\hat{h}}_A^{\mathbb{B}}(P)\ge C_1[K(P):K{]}^{-2}$$ for all points $P\in A(\bar{K})$ whose height satisfies $0<{\hat{h}}_A(P)\le C_2$.

48 pages