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May 25, 2021 at 13:58 vote accept Watson
May 24, 2021 at 17:51 comment added Yuri Zarhin You are welcome. Yes, you understand it correctly: it certainly contradicts. The source of a mistake in the other answer is an improper use of Tate's theorem. Indeed, this theorem describes the local $v$-adic (or $\mathfrak{P}$-)invariants of the division algebra of endomorphisms. While these invariants lie in $Q/Z$, the other answer tacitly (and erroneously) assumes that they are rational numbers.
May 24, 2021 at 16:15 comment added Watson Thank you very much for this extended answer! I am going to read it, but just a quick question: you are considering the geometric endomorphism algebras, right (and not just the endomorphisms defined over $\mathbb F_{p^2}$)? Oh, I just saw that you mention this at the very end of your post. Because then your construction contradicts the other answer, if I understand correctly... !
May 24, 2021 at 15:37 history answered Yuri Zarhin CC BY-SA 4.0