Quotients of Tate modules

Question

Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let $\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over $K$ and denote by $T_{p}A$ its Tate module.

When $A$ has ordinary reduction then it is known that there is an exact sequence of $Gal(\overline{K}:K)$-modules: $$ 0 \rightarrow \mathbb{Z}_{p}(-1) \rightarrow T_{p}A \rightarrow \mathbb{Z}_{p} \rightarrow 0. $$ I would be interested in a characterization of the quotient $\mathbb{Z}_{p}$ and its applicability in the case of supersingular reduction. When $$ HT: H^{1}(A,{\cal O})\otimes_{K} \mathbb{C} _{p}\oplus H^{0}(A,\Omega ^{1})\otimes_{K} \mathbb{C} _{p}(-1) \simeq Hom_{\mathbb{Z} _{p}}(T_{p}A ,\mathbb{C} _{p}). $$ denotes the Hodge-Tate isomorphism, let $M \subset T_{p}A$ be the intersection of all kernels of homomorphisms in the image of $H^{1}(A,{\cal O})\otimes_{K} \mathbb{C} _{p}$ in $Hom_{\mathbb{Z} _{p}}(T_{p}A ,\mathbb{C} _{p})$ under $HT$. It is easy to see that the quotient $T^{Q}_{p}A:= T_{p}A/M$ is a free $\mathbb{Z} _{p}$-module of rank $\leq 2$, and that it gives rise to an exact sequence: $$0 \rightarrow M \rightarrow T_{p}A \rightarrow T^{Q}_{p}A \rightarrow 0. $$ If $A$ has ordinary reduction, then $T^{Q}_{p}A$ may be identified with $\mathbb{Z}_{p}$ from the first diagram.

If $A$ has supersingular reduction, is then $T^{Q}_{p}A$ also a free $\mathbb{Z}_{p}$-module of rank one? More generally, the quotient $T_{p}^{Q}A$ could be also defined when $A$ is an abelian variety of dimension $g$. Will $T_{p}^{Q}A$ be a free $\mathbb{Z}_{p}$-module of rank $g$?

Answer 1

Although this question isn't really well-defined (you'd surely need to be more precise about the word "canonical" in the comment under the question) let me make two comments which hopefully put this to bed. Firstly it's hard to see how a "canonical" map from $T_pA$ would not have Galois-stable kernel -- because if the kernel were not stable then just hit it with Galois and you've got a different, but equally "canonical", map, so, as David Loeffler pointed out in the comments, if the Tate module is irreducible (which it certainly sometimes is: for example with a supersingular elliptic curve over $\mathbf{Q}_p$ I guess even the $p$-torsion is irreducible) then you're already in trouble. "Galois-stable" is just a more concrete say of saying "only depends on things up to isomorphism", which surely you want any "canonical" construction to have.

But here's another way of looking at it (which is probably just the same way in disguise). Let me argue that it is unreasonable even to hope for a "canonical" map mod $p$, i.e. a map from $A[p]$ to a 1-dimensional space over $\mathbf{F}_p$. This time I'll interpret "canonical" as "behaves well in families" -- because if $A$ is a family of ordinary elliptic curves then there is a family of subgroups of order $p$ which at each fibre is the so-called "canonical subgroup", and there's your kernel. However such a family cannot possibly exist for a general family which includes non-ordinary curves, because if it did then you would be constructing a section of the forgetful functor $Y_0(p)\to Y_0(1)$ over $\mathbf{Q}_p$, and such a section does not exist [to make this completely rigorous you could add a point of order $N$ everywhere to make the functors representable, suppress the point from the notation, and then just note that $Y_0(p)$ is now a connected curve with bigger genus than $Y_0(1)$]. The fact that you can write down such a section on the ordinary locus, first noticed by Katz I guess, is exactly the mod $p$ version of your question, and so even mod $p$ it's hard to see you moving far into the non-ordinary locus. You can move a little way in -- this is the theory of the canonical subgroup -- but the moment you're a little way in there are only canonical subgroups of order $p^n$ for finitely many $n$ so you'll never glue to get a map on the whole Tate module.

Answer 2

Thank you for your answers.

After thinking some time about it, it seems that such a map as in my comment above cannot even exist in the case where $A$ has super singular reduction.

If there were a surjection $f:T_{p}A \rightarrow T_{p}^{Q}A$ of $\mathbb{Z}_{p}$-modules as described in my comment above, then the kernel $M'$ of $f$ would be a non-zero by dimension considerations.

Since by assumtion the image of $Hom_{\mathbb{Z}_{p}}(T_{p}^{Q}A,\mathbb{C}_{p})$ in $Hom_{\mathbb{Z}_{p}}(T_{p}A,\mathbb{C}_{p})$ coincides with $HT(H^{1}(A,{\mathcal O})\otimes _{K}\mathbb{C}_{p})$, the module $M'$ must be a submodule of $M$, which was defined as the intersection of all kernels of homomorphisms in $HT(H^{1}(A,{\mathcal O})\otimes _{K}\mathbb{C}_{p}) \subset Hom_{\mathbb{Z}_{p}}(T_{p}A,\mathbb{C}_{p})$.

As $M$ must be irreducible as a Galois module, as remaked by David, and since it has a nontrivial submodule $M'$, the module $M$ must be equal to $T_{p}A$. Therefore, all homomorphisms in $HT(H^{1}(A,{\mathcal O})\otimes _{K}\mathbb{C}_{p})$ must be trivial. And this cannot be since it a $\mathbb{C}_{p}$-vector space of dimension one.

Is this correct or am I missing something?

Answer 3

This is not an answer, but a clarification about a typo in the question. The question says that if $p$ is a prime of good ordinary reduction for an elliptic curve $E$, then we have an exact sequence of $\text{Gal}(\overline{K}/K)$-modules $$0 \to \mathbf{Z}_p(-1) \to T_pE \to \mathbf{Z}_p \to 0.$$ However, the above exact sequence is not an exact sequence of $\text{Gal}(\overline{K}/K)$-modules. It is in fact an exact sequence of $\text{Gal}(\overline{K}/\hat{K^{ur}})$-modules, where $\hat{K^{ur}}$ is the (completion of) the maximal unramified extension of $K$.

So what does $T_pE$ look like as a $\text{Gal}(\overline{K}/K)$-module? Here, unfortunately, we can say far less. Assume that $p$ is a prime of good ordinary reduction for $E$. Then all we know is that there is an exact sequence of $\text{Gal}(\overline{K}/K)$-modules $$0 \to T^{'} \to T_pE \to T^{''} \to 0$$ where $T^{'}$ and $T^{''}(-1)$ are unramified as $\text{Gal}(\overline{K}/K)$-modules*. I don't think we can say what $T^{'}$ and $T^{''}$ look like explicitly as $\text{Gal}(\overline{K}/K)$-modules because that would depend on the specific curve $E$.

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*This fact from Kato's Asterisque 295 paper "$p$-adic hodge theory and zeta values of modular forms", Proposition 17.1, part (iii). Set $T^{'}$ and $T^{''}$ as in Proposition 17.2.