A Lehmer-Type Lower Bound for the Canonical Height on Elliptic Curves Over Function Fields

2024/02/22 by Silverman, Joseph H.
#14G40 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11G05 #Secondary: 11R58

paper · doi:10.48550/arxiv.2402.14771

Abstract

Let \mathbbF be the function field of a curve over an algebraically closed field with char(\mathbbF)≠2,3, and let E/\mathbbF be an elliptic curve. Then for all finite extensions \mathbbK/\mathbbF and all non-torsion points P∈E(\mathbbK), the \mathbbF-normalized canonical height of P is bounded below by h_E(P) ≥ \frac110500⋅ h_\mathbbF(j_E)²⋅ [\mathbbK:\mathbbF]².

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