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

Looper, Nicole; Silverman, Joseph

doi:10.1090/tran/9024

A Lehmer-type height lower bound for abelian surfaces over function fields
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by Nicole R. Looper and Joseph H. Silverman;

Trans. Amer. Math. Soc. 377 (2024), 1915-1955

DOI: https://doi.org/10.1090/tran/9024

Published electronically: December 12, 2023

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Abstract:

Let $K$ be an 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 $A(\overline {K})$. 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 $P\in {A(\overline {K})}$ whose height satisfies $0<\hat {h}_A(P)\le {C_2}$.

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