Specialization of canonical heights on abelian varieties

2021/10/14 by Alexander Carney, Carney, Alexander
Biochemistry, Genetics and Molecular Biology · Mathematics · Social Sciences · #11G10 #11G50 #14G40 #37P30 #Algebraic Geometry and Number Theory #FOS: Mathematics #Ginseng Biological Effects and Applications #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2110.07664

openalex publication_date 2021/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a family of abelian varieties over a quasiprojective smooth curve T⁰ over a global field and a point P on the generic fiber, we show that the Néron-Tate canonical height h_Xt(P_t) of P_t along each fiber is exactly equal to a Weil height h _M(t) given by an adelic metrized line bundle M on the unique smooth projective curve T containing T⁰. As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of P is equivalent to M being big.

Citations

• Heights and arithmetic dynamics over finitely generated fields

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