TITLE: Relation - Anabelian geometry and Tate conjecture
QUESTION [5 upvotes]: A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture.

I would like to know what is the relation between Anabelian algebraic geometry and Tate conjecture.

Grothendieck said (Esquisse d'un programme): "c'est alors que se dégage la 'conjecture fondamentale de la géométrie algébrique anabélienne', proche des conjectures de Mordell et de Tate que vient de démontrer Faltings"

REPLY [5 votes]: The Tate conjecture is the statement that $End(A,B)\otimes \mathbb{Q}_p$ is isomorphic to $End_G(T_p A,T_p B)\otimes \mathbb{Q}$ for abelian varieties $A,B$ over a number field $K$ with absolute Galois group $G$ and $T_pA$ the Tate module of $A$. A map $\pi_1(A) \to \pi_1(B)$ preserving the fundamental exact sequences $1 \to \pi_1({\bar A}) \to \pi_1(A) \to G \to 1$, etc, is a map $\pi_1({\bar A}) \to \pi_1({\bar B})$ preserving the Galois action and since $\pi_1({\bar A}) = \prod_p T_pA$, we can see that the Tate conjecture is pretty close to a statement that map between the $\pi_1$'s correspond to maps between the varieties. I think it's not quite the same because of the $\mathbb{Q}$ coefficients, but abelian varieties are not anabelian :-).