A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture.



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



Relation - Anabelian geometry and Tate conjecture
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 :-). 


