Explicit bounds on torsion of CM abelian varieties over \(p\)-adic fields with values in Lubin-Tate extensions (Q6579960)

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).