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

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 $L=K({\mu }_{p^{\infty }})$). In this paper, 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 $p$-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.

arXiv comment21 pages, some minor modifications

References in corpus (1)

• Torsion of rational elliptic curves over the maximal abelian extension of Q