# Status of Grothendieck's conjecture on homomorphisms of abelian schemes

In [1] Grothendieck posits the following:

Conjecture. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian schemes over $S$, $l$ a prime number, $u_l: T_l(A) \rightarrow T_l(B)$ a homomorphism, and suppose that there exists a point $s\in S$ such that $u_{ls}$ comes from a homormophism $u_s: A_s \rightarrow B_s$ of abelian schemes over $k(s)$. Then there exists an integer $n>0$ and a homomorphism $v: A \rightarrow B$ such that $T_l(v)=n u_l$.

Next Grothendieck remarks that the conjecture follows from Tate's conjectures on algebraic cycles. He then proceeds to a give a proof, using the Serre-Tate theorem, for the case of a characteristic zero base field for which the statement holds with $n=1$.

Question: What is the current status of this conjecture? If not resolved, are there any partial results for the positive characteristic case?

Here is what I've spotted in the literature so far, all characteristic zero:

Deligne in the second of three papers on Hodge theory, proves results about homomorphisms of abelian schemes over schemes $S$ of finite type over $\mathbf{C}$. For instance:

Proposition (Deligne 4.4.12 [2]) : Let $f: X \rightarrow S$ be an abelian scheme over a smooth scheme $S$. The following conditions are equivalent:

$(i)$ For every abelian scheme $g: Y \rightarrow S$, we have $$\mathrm{Hom}_S(X,Y) \cong \mathrm{Hom}_S(R_1 f_* \mathbf{Z}, R_1 g_* \mathbf{Z} )$$

$(ii)$ The condition $(i)$ is verified for $X=Y$, and the centre $Z$ of $\mathrm{End}_S(X)\otimes \mathbf{Q}$ does not admit a complex place $\rho: Z \rightarrow \mathbf{C}$ such that the direct factor $R_1 f_*\mathbf{Q} \otimes_{Z,\rho} \mathbf{C}$ of $R_1 f_* \mathbf{C}$ is of pure Hodge type (-1,0).

Along similar lines, S.G. Tankeev in a 1976 paper [3] proves if $S$ is a connected smooth curve over $\mathbf{C}$, and $\pi_i: X_i \rightarrow S, i=1,2$ are two abelian schemes, then under some natural conditions, the canonical homomorphism $$\mathrm{Hom}_S(X_1,X_2) \rightarrow \mathrm{Hom}(R_1 \pi_{1*} \mathbf{Z}, R_1 \pi_{2*} \mathbf{Z})$$ is an isomorphism. In a paper the following year [4] Tankeev proves a Tate module variant of the above, of the form $$\mathrm{Hom}_k(I_1,I_2)\otimes_\mathbf{Z} \mathbf{Z}_l \cong \mathrm{Hom}_G(T_l(I_1), T_l(I_2)),$$ where $G=\mathrm{Gal}(\overline{k}/k)$, and $I_1, I_2$ are abelian varieties over $k(t)$ whose Néron models admit compactifications with certain properties.

Are there other well-known results along the same lines? I would especially like to know about the characteristic $p$ case, as well any results that actually spread a homomorphism of abelian varities over a point of S to maps of abelian schemes over S, as in the conjecture.

[1] A. Grothendieck Un théorème sur les homomorphismes de schémas abéliens Invent. Math. 2 59-78 (1966)

[2] P. Deligne Théorie de Hodge: II Publications mathématiques de l'I.H.E.S, tome 40 (1971) p.5-57

[3] S.G. Tankeev On homomorphisms of abelian schemes Math. USSR Izvestija Vol. 10 (1976), No. 4

[4] S.G. Tankeev On homomorphisms of abelian schemes II Math. USSR Izvestija Vol. 11 (1977), No. 6

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Suppose that $S$ is a normal variety over $\mathbb{F}_p$ with generic point $\eta\to S$. Then, by Zarhin, there exists a homomorphism of the generic fibers $u_{\eta}:A_{\eta}\to B_{\eta}$ giving rise to $u_{\ell}$ (one doesn't even need the existence of $u_s$ for this). Now, by Proposition 2.7 in Chai-Faltings 'Degeneration of abelian varieties', $u_{\eta}$ extends over all of $S$. The idea basically is to use the Neronian property of abelian schemes to show that the closure of the graph of $u_{\eta}$ in $A\times B$ maps isomorphically onto $A$. – Keerthi Madapusi Pera Mar 1 '13 at 14:39
Thanks, this is great! Though I'm somewhat confused about the application of Zarhin to obtain $u_\eta$. Isn't $Hom_G(T_l(A), T_l(B))$ only isomorphic to $Hom(A,B)$ up to tensoring with $\mathbf{Z}_l$? – Zavosh Mar 1 '13 at 19:25
Ah, that's right! I was being cavalier as usual. I think the existence of $u_s$ should force it to actually lie in $Hom(A_{\eta},B_{\eta})$. – Keerthi Madapusi Pera Mar 1 '13 at 20:45
More precisely, what my comment shows is that we have $Hom(A_{\eta},B_{\eta})=Hom(A,B)$. So, by Zarhin, $u_{\ell}$ arises from an element of $Hom(A,B)\otimes\mathbb{Z}_{\ell}$. To check that some multiple of it actually lies in $Hom(A,B)$, it suffices to check that, for some point $s\in S$, the image in $Hom(A_s,B_s)\otimes\mathbb{Z}_{\ell}$ lies in $Hom(A_s,B_s)$. – Keerthi Madapusi Pera Mar 1 '13 at 20:50
That sounds right. If you would like to enter it as an answer I would accept it. – Zavosh Mar 2 '13 at 0:17

This is an expansion of my comments. Grothendieck's conjecture is now a theorem, except perhaps when $S$ is in characteristic $p$ and $\ell=p$. The explanation that follows is presumably what Grothendieck had in mind in his comment regarding the Tate conjecture.
First, we make some reductions: We can certainly assume that $S$ is integral. In fact, we can assume that $S$ is normal. This follows from Proposition 1.2 of Grothendieck's article.
Finally, we can assume that the generic point $\eta$ of $S$ is the spectrum of a finitely generated field. This can be shown as in 2.2 of Grothendieck's article.
In this scenario, by Proposition 2.7 in Chai-Faltings, 'Degeneration of Abelian varieties', the natural restriction map $Hom(A,B)\to Hom(A_{\eta},B_{\eta})$ is a bijection. Suppose now that $\eta$ is the spectrum of a finitely generated field. Then, by the Tate conjecture for homomorphisms of abelian varieties over finitely generated fields (proved by Zarhin in finite characteristic, and Faltings in characteristic $0$), the natural map of $\mathbb{Z}_{\ell}$-modules: $$Hom(A,B)\otimes\mathbb{Z}_{\ell}\to Hom_G(T_{\ell}(A_{\eta}),T_{\ell}(B_{\eta}))$$ is an isomorphism. Here, on the right hand side, $G$ is the absolute Galois group of the spectrum of $\eta$. In particular, $u_{\ell}$ gives rise to an element of $Hom(A,B)\otimes\mathbb{Z}_{\ell}$.
Now, suppose that the specialization of $u_{\ell}:T_{\ell}(A)\to T_{\ell}(B)$ over a point $s\in S$ arises from an honest homomorphism $u_s:A_s\to B_s$. Then we find that, for some $n\in\mathbb{Z}_{\geq 1}$, $nu_{\ell}$ arises from a homomorphism $u':A\to B$. Essentially, the map $$Hom(A,B)_{\mathbb{Q}}\to Hom(A_s,B_s)_{\mathbb{Q}}$$ identifies the left hand side with a vector sub-space of the right hand side. In fact, if $S$ is of characteristic $p$, we can take $n$ to be a power of $p$, since the specialization map will have saturated image after inverting $p$.