Explicit bounds on torsion of CM abelian varieties over \(p\)-adic fields with values in Lubin-Tate extensions
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 \(L = K(\mu_{p^{\infty}})\)). The author shows that there exists an explicitly described constant \(C\), depending only on \(K\), \(k\) and an integer \(g \geq 1\), 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 \(C\) (Theorem 3.1). Similar bound is given in the case where \(L = K(K^{1/p^{\infty}})\). Main results are next applied to study bounds of orders of torsion subgroups of some \(CM\) abelian varieties over number fields with values in full cyclotomic fields (Theorem 1.3).
scientific article; zbMATH DE number 7887958
scientific article