@nfdc23 Ah, yes, I see. I was being silly! You're just saying that $T_p A=T_p \mathscr{A}$, where $\mathscr{A}$ is a model, and that we get a decomposition into an étale part a twist. Why is it not $M(1)$ though? Oh, because you wrote it cohomologically. OK, so I feel satisfies as soon as someone can come and verify my reduction. Thanks again nfdc23!

Surely the Arthur-Selberg trace formula is an example of such a theorem.

@nfdc23 Also, unless I'm being terribly silly (and I might) won't the canonical lift's endomorphism algebra contain the lift of Frobenius—how can it then be $\mathbb{Q}_p$? Finally, do you know conditions on $A$ that will make this true?

@nfdc23 Hello nfdc23, thanks for your comment. A few questions. First, it seems as though you're answering the $\ell=p$ Tate's isogeny theorem part of my question—you agree then that this is an equivalent formulation of the original statement? Second, I'd appreciate if you could expand upon your comment. Why does the generic fiber of the canonical lift have this cohomology necessarily? I looked in some articles (e.g. Katz) and couldn't find anything.

Thank you for your nice answer. :) I will certainly think hard on it!

@pbelmans Thank you for your comment! I was aware of this actually, and wanted actual locally ringed space. The intuition you gave is roughly the intuition I have as well.