*Please see edits below!*

So, let $A,A'/K$ be abelian varieties where $K$ is a $p$-adic local field with residue field $k$. Suppose further that they have good reduction with models $\mathscr{A},\mathscr{A}'/\mathcal{O}_K$. I think it follows from Serre-Tate theory that

$$\text{Hom}(A,A')=\left\{g\in\text{Hom}(\mathscr{A}_k,\mathscr{A}'_k):g^\ast\text{ preserves Hodge filtration}\right\}$$

where

$$g^\ast:H^1_\text{crys}(\mathscr{A}'_k/W(k))\to H^1_\text{crys}(\mathscr{A}/W(k))$$

is the obvious map.

In particular, we get a factorization

$$\text{Hom}(A,A')\hookrightarrow \text{Hom}(\mathscr{A}_k,\mathscr{A}'_k)\hookrightarrow \text{Hom}_{F,V}(H^1_\text{crys}(\mathscr{A}'/W(k)),H^1_\text{crys}(\mathscr{A}_k/W(k))$$

Now, it's the famous result of Tate that the induced map

$$\text{Hom}(\mathscr{A}_k,\mathscr{A}'_k)\otimes_\mathbb{Z}\mathbb{Z}_p\to \text{Hom}_{F,V}(H^1_\text{crys}(\mathscr{A}'/W(k)),H^1_\text{crys}(\mathscr{A}_k/W(k)))$$

is an isomorphism. My hope is then that the injection

$$\text{Hom}(A,A')\otimes_\mathbb{Z}\mathbb{Z}_p \hookrightarrow \text{Hom}_{F,V}(H^1_\text{crys}(\mathscr{A}'_k/W(k)),H^1_\text{crys}(\mathscr{A}_k/W(k)))$$

has image precisely those $F,V$-commuting maps

$$H^1_\text{crys}(\mathscr{A}'_k/W(k))\to H^1_\text{crys}(\mathscr{A}/W(k))$$

that preserve the Hodge filtration.

But, this is not obvious to me. The examples I've computed seem to work. This is also the desired analogue of what happens over $\mathbb{C}$ where

$$\text{Hom}(A,A')=\left\{g\in\text{Hom}(H^1_\text{sing}((A')^\text{an},\mathbb{Z}),H^1_\text{sing}(A^\text{an},\mathbb{Z}):g_\mathbb{C}\text{ preserves Hodge filtration}\right\}$$

which, again, is motivation but not proof.

I think that via the fully faithfulness of $D_\text{crys}$ that this may be related to an $\ell=p$ version of Tate's conjecture for $p$-adic fields. Specifically, let's restrict to the case when $K=\mathbb{Q}_p$. Then, the fully faithfulness of $D_\text{crys}$ should say that

$$\text{Hom}_{\mathbb{Q}_p[G_{\mathbb{Q}_p}]}(H^1_{\acute{e}\text{t}}(A'_{\overline{K}},\mathbb{Q}_p),H^1_{\acute{e}\text{t}}(A_{\overline{K}},\mathbb{Q}_p))\xrightarrow{\approx}\text{Hom}_{F,V,\text{Fil}}(H^1_\text{crys}(\mathscr{A}'_{\mathbb{F}_p}/\mathbb{Q}_p),H^1_\text{crys}(\mathscr{A}_{\mathbb{F}_p}/\mathbb{Q}_p))$$

and thus it seems as though asking whether

$$\text{Hom}(A,A')\otimes_\mathbb{Z}\mathbb{Q}_p\to \text{Hom}_{F,V,\text{Fil}}(H^1_\text{crys}(\mathscr{A}'_{\mathbb{F}_p}/\mathbb{Z}_p),H^1_\text{crys}(\mathscr{A}_{\mathbb{F}_p}/\mathbb{Z}_p))\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$$

is an isomorphism amounts essentially to whether

$$\text{Hom}(A,A')\otimes_\mathbb{Z}\mathbb{Q}_p\to \text{Hom}_{\mathbb{Q}_p[G_{\mathbb{Q}_p}]}(H^1_{\acute{e}\text{t}}(A'_{\overline{K}},\mathbb{Q}_p),H^1_{\acute{e}\text{t}}(A_{\overline{K}},\mathbb{Q}_p))$$

is an isomorphism which, of course, is something like Tate's isogeny conjecture for $\ell=p$.

Any help is greatly appreciated!

**EDIT:** Assuming that I've correctly applied the fully faithfuless of $D_\text{crys}$ above, and my question does imply Tate's isogeny theorem for $\ell=p$ for $p$-adic fields, then this question (as well, perhaps, as the comments below) give counterexamples to the claim.

So, could someone just verify that my deduction of Tate's isogeny theorem from an affirmative answer to my original question is correct? Thanks!

**EDIT EDIT:** I am now confident that I didn't err in my transformation of the original question into Tate's isogeny conjecture. So, unless someone would like to give an answer to the question adding something else of interest, I'll just answer my question as a community wiki. Thanks!

finite fieldsgives equality (i.e., affirmative answer!) for ordinary canonical lifts. $\endgroup$ – nfdc23 Apr 29 '16 at 13:47