If A is an abelian variety over a p-adic field K, then it is a theorem of Mattuck that the torsion subgroup of the Mordell-Weil group A(K) is finite. It is an interesting theorem of Kubo and Taguchi in 2013 that the torsion subgroup of A(L) is also finite if A has potential good reduction and L is the extension field of K obtained by adjoining all p-power roots of all elements of K. In this talk, we will discuss a “uniform version” of Kubo-Taguchi for CM abelian varieties. The main theorem is as follows: There exists a constant C, depending only on K and dim A, such that the order of the torsion subgroup of A(L) is bounded by C under the assumption that A has complex multiplication.

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