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?
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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).




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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. 




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[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



Status of Grothendieck's conjecture on homomorphisms of abelian schemes
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 $p$.


