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The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked.

Let $p$ be a prime number, and $\mathcal{C}$ an $\mathbf{F}_p$-isogeny class of $\mathbf{F}_p$-simple abelian varieties of dimension $g$ associated with a certain Weil $p$-number $\pi$ which will be assumed not real, i.e., $\pi\neq\pm\sqrt{p}$. For any $A$ in $\mathcal{C}$, the $\mathbf{Q}$-algebra $\textrm{End}_{\mathbf{F}_p}(A)\otimes\mathbf{Q}$ is commutative and equal to the non-real CM number field $\mathbf{Q}(\pi)$, which has degree $2g$. The endomorphism ring of any object of $\mathcal{C}$ is an order of this field containing $\pi$ and $p/\pi$. I would like to assume, for simplicity, that

(*) the order $\mathbf{Z}[\pi, p/\pi]$ is maximal at $p$.

Let $A$ be a fixed object of $\mathcal{C}$, and let $R_A$ be its endomorphism ring. To any nonzero ideal $I$ of $R_A$ we can attach, following Waterhouse, the finite $\mathbf{F}_p$-subgroup $H(I)$ of $A$ consisting of the (subgroup-scheme theoretic) intersection of all kernels of the nonzero elements of $I$.

On the other hand, associated to any finite $\mathbf{F}_p$-subgroup $N$ of $A$ there is the ideal $J(N)$ of $R_A$ consisting of all morphisms $\varphi:A\to A$ vanishing on $N$.

Waterhouse calls a nonzero ideal $I$ of $R_A$ a kernel ideal for $A$ if the equality $I=J(H(I))$ holds.

The question I have asks for a correspondence between ideals of $R_A$ and finite subgroups of $A$ defined over $\mathbf{F}_p$:

Q: Can we find $A$ such that for all nonzero ideals $I$ of $R_A$ and for all finite $\mathbf{F}_p$-subgroups $N$ of $A$ we have $I=J(H(I))$ and $N=H(J(N))$?

The idea behind the question is to ask for an abelian variety $A$ such that the category $\mathcal{C}$ can be reconstructed starting from the contravariant functor $T\mapsto h_A(T)=\mathrm{Hom}_\mathcal{C}(T,A)$ going from $\mathcal{C}$ to rank-one, torsion free $R_A$-modules.

If I got it right, question Q is equivalent to ask $h_A$ be an anti-equivalence of categories. (At least if Q holds for a certain object $A$ then the equality $I=J(H(I))$ ensures essential surjectivity of $h_A$ and the assignment $I\mapsto A/H(I)$ gives an inverse functor defined on nonzero ideals of $R_A$).

Waterhouse proves several facts related to Q (which led to me ask the question to begin with):

1) Let $I$ and $J$ be kernel ideals for $A$. Then $A/H(I)\simeq A/H(J)$ in $\mathcal{C}$ if and only if $I\simeq J$ as $R_A$-modules;

2) If $I$ is a kernel ideal, then $\textrm{End}_{\mathbf{F}_p}(A/H(I))$ is the order of $I$ (inside $\mathbf{Q}(\pi)$);

3) If the $\ell$-adic Tate module $T_\ell(A)$ is free (of rank one) over $R_A\otimes\mathbf{Z}_\ell$ for all primes $\neq\ell$ and the Dieudonn\'e module $T_p(A)$ is free over $R_A\otimes\mathbf{Z}_p$, then every nonzero ideal of $R_A$ is a kernel ideal for $A$;

(the assumption (*) above ensure freeness of $T_p(A)$, if I'm right)

4) Any $A'$ in $\mathcal{C}$ is isogenous to an $A$ as in 3) in such a way that $R_{A'}\simeq R_A$ (the isomorphism between the two rings being definable from the isogeny from $A'$ to $A$);

5) Any order of $\mathbf{Q}(\pi)$ containing $\pi$ and $p/\pi$ arises as endomorphism ring $R_A$, for some $A$.

These results suggest that in order to find a good candidate for question Q we should start with an object $A$ such that $R_A$ is $\mathbf{Z}[\pi,p/\pi]$ and such that $T_\ell(A)$ is free over $R_A\otimes\mathbf{Z}_\ell$ for any $\ell$. It follows from 3), 4) and 5) that such an $A$ exists.

Since by 3) we have $I=J(H(I))$, we see that Q has a positive answer if for any $\mathbf{F}_p$-subgroup $N$ of $A$ we have $N=H(J(N))$. This equality holds if and only if for all primes $\ell$ the $\ell$-primary parts $N_\ell$ and $H(J(N))_\ell$ of the two subgroups coincide. If $\ell\neq p$, then the equality $N_\ell=H(J(N))_\ell$ amounts to the following:

Let $\Lambda$ be the $R_A\otimes\mathbf{Z}_\ell$-stable lattice of $V_\ell(A):=T_\ell(A)\otimes\mathbf{Q}_\ell$ containing $T_\ell(A)$ and corresponding to $N_\ell$.

Q($\ell$): Is it true that $\Lambda$ is equal to the intersection of all pre-images of $T_\ell(A)$ under the $R_A\otimes\mathbf{Z}_\ell$-endomorphisms of $V_\ell(A)$ sending $\Lambda$ to $T_\ell(A)$? Equivalently, is it true that for any $v\in V_\ell(A)$ with $v\notin\Lambda$ there exists an $R_A\otimes\mathbf{Z}_\ell$ lattice $\Lambda'\subset V_\ell(A)$ containing $\Lambda$, not containing $v$, and isomorphic to $T_\ell(A)$?

Since the assumption (*) ensures that $N_p=H(J(N))_p$, Q has a positive answer if and only if Q($\ell$) does for a single object $A$ for all $\ell\neq p$.

Question Q can be tested on ordinary $\mathcal{C}$ using a result of Deligne (Variétés abéliennes ordinaires sur un corps fini. Invent. Math. 8 1969 238–243) which says, when applied to the isogeny class $\mathcal{C}$ (and using the maximality assumption (*)), that $\mathcal{C}$ is equivalent to the category of rank one, torsion free $\mathbf{Z}[\pi,p/\pi]$-modules. The proof of this elegant result uses Serre-Tate canonical lifting of ordinary AVs. Using this theorem, Q translates into this other question, where $R$ should be taken equal to $\mathbf{Z}[\pi,p/\pi]$.

If you have anything to say about Q or Q($\ell$), or both, I would be happy to hear it.

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