Explicit bounds on torsion of CM abelian varieties over $p$-adic fields with values in Lubin–Tate extensions

Yoshiyasu Ozeki

Vol. 330 (2024), No. 1, 171–197

DOI: 10.2140/pjm.2024.330.171

Abstract

Let $K$ and $k$ be $p$-adic fields. Let $L$ be the composite field of $K$ and a certain Lubin–Tate extension over $k$ (including the case where $L=K({\mu }_{p^{\infty }})$). We show that there exists an explicitly described constant $C$, depending only on $K$, $k$ and an integer $g\ge 1$, which satisfies the following property: if $A_{∕K}$ is a $g$-dimensional CM abelian variety, then the order of the $p$-primary torsion subgroup of $A(L)$ is bounded by $C$. We also give a similar bound in the case where $L=K(\sqrt[p^{\infty }]{K})$. Applying our results, we study bounds of orders of torsion subgroups of some CM abelian varieties over number fields with values in full cyclotomic fields.

Keywords

abelian varieties, Lubin–Tate extensions

Mathematical Subject Classification

Primary: 11G10

Milestones

Received: 23 October 2023

Revised: 16 May 2024

Accepted: 8 June 2024

Published: 22 July 2024