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Let $K$ be a finitely generated extension of an algebraically closed field of characteristic zero, and $A,B$ abelian varieties over $K$.

Then is $Hom_K(A,B)\otimes \mathbb{Z_l} \cong Hom_{Gal(\bar{K} / K)}(T_l(A),T_l(B))$, where $T_l(A)$ is the $l$-adic Tate module of $A$?

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This might help,… – Matt Mar 12 '13 at 19:02
up vote 10 down vote accepted

The answer is no: an easy (not interesting) counterexample is provided by "constant" abelian varieties, i.e., when $A$ and $B$ are defined over an algebraically closed field $k$ of characteristic zero while $K$ is finitely generated over $k$.

So, let's assume that the $K/k$-traces of $A$ and $B$ are zero, i.e., both $A$ and $B$ do not contain constant abelian subvarieties of positive dimension. Actually, we need more: assume that there are no {\sl isotrivial} abelian subvarieties of positive dimension, i.e., there are no abelian subvarieties (except zero) that become constant after a finite algebraic extension of $K$.

Let us assume also that $k$ is the field $C$ of complex numbers. Then $A$ and $B$ become generic fibers of abelian schemes $\mathcal{A}$ and $\mathcal{B}$ over a smooth quasiprojective complex algebraic variety $S$ and the analogue of Tate's conjecture becomes equivalent to a similar question about homomorphisms between the first integral homology groups $H_1(\mathcal{A}_s,Z)$ and $H_1(\mathcal{B}_s,Z)$ of the fibers that commute with the actions(s) of the fundamental group $\pi_1(S,s)$ of the base $S$. Here $s$ is a complex point of $s$ while the corresponding fibers $\mathcal{A}_s$ and $\mathcal{B}_s$ are complex abelian varieties and the question is whether all $\pi_1(S,s)$-equivariant homomorphisms between $H_1(\mathcal{A}_s,Z)$ and $H_1(\mathcal{B}_s,Z)$ come from homomorphisms of abelian varieties $\mathcal{A}_s \to \mathcal{B}_s$?

If either $\dim(A)\le 3$ or $\dim(B)\le 3$ then the answer is yes: see Section 4.4 of Delignes's Th\'eorie de Hodge.II 40/PMIHES_1971_40_5_0/PMIHES_1971_40_5_0.pdf"> However, there is a counterexample with $4$-dimensional $A=B$ (Faltings, Inv. Math. 73(1983), 337-347). See also arXiv:math/0504523 [math.AG] .

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