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Experimental investigations on Lehmer's conjecture for elliptic curves

Sven Cats, John Michael Clark, Charlotte Dombrowsky, Mar Curco Iranzo, Krystal Maughan, Eli Orvis

TL;DR

This work tackles Lehmer’s conjecture for elliptic curves by computing the smallest possible canonical height $\hat{h}(P)$ of a non-torsion point defined over extensions of fixed degree, formalized as $C_E = \inf \{ \hat{h}(P) [K(P):K] \}$. It also considers Lang’s refinement $C_{K,d}$ using a curve-invariant bound $M_E = \max\{ h(j_E), \log |N_{K/\mathbb{Q}}\Delta_E|, 1 \}$ and $C_{K,d} = \inf\{ \hat{h}(P)/M_E \}$ for degree-$d$ extensions. The authors develop a finite-search framework by reducing to a finite subfamily $\mathscr{F}'$ via a discriminant-bound $\Delta(D,E,F)$ and a refined height bound $h(P)-\hat{h}(P)$ (a modified CPS bound), enabling practical computation of minimal-height points over quadratic fields. Applying this to $17{,}834$ curves over $\mathbb{Q}$, they obtain 86 curves with provably smallest quadratic-field points and provide detailed field-defining data, observing limited correlation with classical invariants and offering empirical guidance for Lehmer–Lang-type conjectures and future computational directions.

Abstract

In this short note, we give a method for computing a non-torsion point of smallest canonical height on a given elliptic curve $E/\mathbb{Q}$ over all number fields of a fixed degree. We then describe data collected using this method, and investigate related conjectures of Lehmer and Lang using these data.

Experimental investigations on Lehmer's conjecture for elliptic curves

TL;DR

This work tackles Lehmer’s conjecture for elliptic curves by computing the smallest possible canonical height of a non-torsion point defined over extensions of fixed degree, formalized as . It also considers Lang’s refinement using a curve-invariant bound and for degree- extensions. The authors develop a finite-search framework by reducing to a finite subfamily via a discriminant-bound and a refined height bound (a modified CPS bound), enabling practical computation of minimal-height points over quadratic fields. Applying this to curves over , they obtain 86 curves with provably smallest quadratic-field points and provide detailed field-defining data, observing limited correlation with classical invariants and offering empirical guidance for Lehmer–Lang-type conjectures and future computational directions.

Abstract

In this short note, we give a method for computing a non-torsion point of smallest canonical height on a given elliptic curve over all number fields of a fixed degree. We then describe data collected using this method, and investigate related conjectures of Lehmer and Lang using these data.

Paper Structure

This paper contains 6 sections, 2 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Computing minimal heights over field extensions
  3. A modified CPS height bound
  4. Computational Results
  5. Description of data
  6. Remarks on Conjectures \ref{['conj:Lehmer']} and \ref{['conj:Lang']}

Key Result

Lemma 2.3

Let $D \in \mathds{R}_{\geq 0}$, $F \in \mathscr{F}$, and $d=[F:K]$. Let $\delta_K$ be the number of Archimedean places of $K$. Define $\Delta(D,E,F) \in \mathds{R}_{>0}$ by If the discriminant $\Delta_F$ of $F$ satisfies $|\Delta_F| \geq \Delta(D,E,F)$, then $\hat{h}(P) \geq \frac{D}{d}$ for all $P \in E(F)-E(F)_{\textnormal{tors}}$ satisfying $K(P) = F$. Further, if $[F:K] = [F' : K]$, then $\D

Theorems & Definitions (10)

  • Conjecture 1.1: Lehmer
  • Conjecture 1.2: Lang
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 3.1