Definition
Given an abelian group A and a prime number p, the p-adic Tate module of A is
where A is the pn torsion of A (i.e. the kernel of the multiplication-by-pn map), and the inverse limit is over positive integers n with transition morphisms given by the multiplication-by-p map A → A. Thus, the Tate module encodes all the p-power torsion of A. It is equipped with the structure of a Zp-module via
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