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