I am reading the article "On the Order of the Reduction of a Point on an Abelian Variety", R.Pink (here is a link to the article).
I am stuck on the proof of Proposition 1.4. Here, the author uses the Hodge-Tate decomposition but I do not understand which decomposition he used (he cited a letter of Serre, but I can not find it anywhere). After searching, the Hodge-Tate decomposition seems to involve cohomology of abelian varieties, while here it seems that the decomposition is used for tori. I don't see how the author uses the decomposition in this situation.
Can someone give me a hint or a reference to the decomposition in the proof?