It is well known that if $A/k$ is an Abelian Variety over (the spectrum of) a field, a very important object to consider is its Tate module $T(A):=\underset{\underset{n}{\longleftarrow}}{\lim}A[p^n](\overline{k})$. Let me assume that the field $k$ is perfect. This object comes equipped with the action of the Galois group of $k$ and it is a free $\mathbb{Z}_p$-module of rank depending on the relation between the characteristic of $k$ and the prime $p$. Moreover, at least in the case of elliptic curves, the Tate module gives a lot of informations about the possible isogenies between two elliptic curves. In fact it is even been proved that, in particular cases, the possible morphisms between Tate modules classify all the possible isogenies between two elliptic curves.

Now, consider $S$ an arbitrary scheme, and let $E/S$ be a relative elliptic curve, i.e. a smooth proper morphism, whose geometric fibers are true elliptic curves. Again, it is possible to define a relative notion of Tate module, as $T_pE=\underset{\underset{n}{\longleftarrow}}{\lim}E[p^n]$, where now, we simply consider it as an inverse limit of group schemes. If we base change to every geometric point, this notion reduces to the one introduced before. Are there general information about this object? In particular, does it admit an action of the Grothendieck Galois group of $S$? We can assume that $S$ is connected, in order to avoid problems in the choice of the geometric point. Moreover, does this object give any information about the isogenies of the corresponding elliptic curves?


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