Tate module of an elliptic curve.

Question

In the case of an elliptic curve $E$ defined over a field $K$, I know that there is a good definition of the so-called Tate module, for every prime $p$, which is the $\mathbb{Z}_p$-module $T_p(E)=\underset{\underset{n}{\longleftarrow}}{\lim}E[p^n](K^s)$, where $E[p^n]$ is the subscheme of $E$ representing its $p$-torsion, and $K^s$ is the separable closure of $K$. I know that this object satisfies very good properties, e.g. it realizes a representation of the Galois Group of $K$. I also know that, under suitable hypothesis, $T_p(E)$ can be used to describe all the possible isogenies between $E$ and itself (or another elliptic curve). Now, I learned via Katz-Mazur book on Arithmetic moduli of Elliptic Curves, that it is possible to define an elliptic curve over arbitrary base rings, and even schemes. I was wondering, but I didn't find any good reference, wether or not it is possible to define a kind of global Tate module. In fact, consider $R$ a ring, and $E/R$ an elliptic curve, i.e. a proper, smooth scheme over $R$ equipped with a zero section, and whose fibers are elliptic curves in the usual sense. Then it is proved in Katz-Mazur that this object again comes equipped with a Group structure, so, is it is meaningful to consider the $p^n$-torsion for varying $n$. Is it possible to define a kind of relative version of the Tate module? My guess is to define it as a functor $T_p(E)$, which sends any $R$-algebra $A$ to $T_p(E)(A)=\underset{\underset{n}{\longleftarrow}}{\lim}E[p^n](A)$. Clearly this defines a sheaf over the category of $R$-Algebras, being an inverse limit of sheaves. Is it representable? And, is there any beautiful and known result about it? For example situations under which it classifies isogenies of elliptic curves over $R$? Thanks a lot for any suggestion or reference!

Answer 1

I don't know why this was never answered.

Let $f:A\to S$ be an abelian scheme and let $\ell$ be invertible on $S$. Then, you can consider the lisse $\mathbb{Z}_\ell$-sheaf $T_\ell A:=(R^1 f_\ast \mathbb{Z}_\ell)^\vee$. This is the relative analogue of the Tate module. For example, by smooth proper base change we have for every $s\in S$ an isomorphism of $\mathrm{Gal}(\overline{k(s)}/k(s))$-modules

$$(R^1 f_\ast \mathbb{Z}_\ell)^\vee_{\overline{s}}\cong H^1_\mathrm{et}(A_{\overline{k(s)}},\mathbb{Z}_\ell)^\vee\cong T_\ell A_{\overline{k(s)}}$$

Note that if $S$ is connected, then you can fully faithfully identify the lisse $\mathbb{Z}_\ell$-sheaf $T_\ell A$ on $S$ with the continuous $\mathbb{Z}_\ell[\pi_1^\mathrm{et}(S,\overline{s})]$-module $T_\ell A_{\overline{k(s)}}$. Similarly, we can think of the rational analogue of the Tate module $V_\ell A:=(R^1 f_\ast\mathbb{Q}_\ell)^\vee$ which is a lisse $\mathbb{Q}_\ell$-sheaf on $S$. Again, we can fully faithfully identify this object with the continuous $\mathbb{Q}_\ell[\pi_1^\mathrm{et}(S,\overline{s})]$-module $V_\ell A_{k(\overline{s})}$.

I think the kind of desired theorem that you're after is the fact that quasi-isogenies out of $A$ (up to equivalence) can be classified by $\pi_1^\mathrm{et}(S,\overline{s})$-invariant open compact subgroups of $V_\ell A_{\overline{k(s)}}$. Is this what you were interested in, or did you mean something else when you were discussing isogenies?