A Lower Bound for the Canonical Height on Abelian Varieties over Abelian Extensions

2003/12/22 by Matthew Baker, Baker, Matthew, Joseph H. Silverman +2 · 3 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.math/0312393

v2 (23 pages), to appear in Mathematical Research Letters. Revised statement and proof of Proposition 7.3, added Lemma 7.9. Some other small revisions made as well

openalex publication_date 2003/12/22 · arxiv created 2004/04/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Let A be an abelian variety defined over a number field K, and consider the canonical height function attached to a symmetric ample line bundle L on A. We prove that there is a positive lower bound C (depending on A, K, and L) for the canonical height of non-torsion points on A defined over the maximal abelian extension K^ab of K.

Citations

• Good Reduction of Abelian Varieties

Cited by

• Explicit Uniform Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions
• Lower bound for the canonical on Abelian varieties over totally p-adic extensions
• New evidence for Rémond's generalisation of Lehmer's conjecture

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