In this case, the Tate module of A is equipped with an action of the absolute Galois group of K, and it is referred to as the Tate module of G. Given an abelian group A and a prime number p, the p-adic Tate module of A is where A[pn] 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[pn+1] → A[pn].
It is equipped with the structure of a Zp-module via When the abelian group A is the group of roots of unity in a separable closure Ks of K, the p-adic Tate module of A is sometimes referred to as the Tate module (where the choice of p and K are tacitly understood).
Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then Tp(G) is a free module over Zp of rank 2d, where d is the dimension of G.[1] In the other case, it is still free, but the rank may take any value from 0 to d (see for example Hasse–Witt matrix).
In the case where p is not equal to the characteristic of K, the p-adic Tate module of G is the dual of the étale cohomology
[3] Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".
[4] In the case of a Jacobian over a curve C over a finite field k of characteristic prime to p, the Tate module can be identified with the Galois group of the composite extension where
is an extension of k containing all p-power roots of unity and A(p) is the maximal unramified abelian p-extension of
[5] Iwasawa exhibited Tp(K) as a module over the completion Zp[[T]] and this implies a formula for the exponent of p in the order of the class groups Cm of the form The Ferrero–Washington theorem states that μ is zero.