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"

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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 :-).

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    $\begingroup$ Can you say a word about the equality $\pi_1(A)=\prod T_pA$? I guess it's tantamount to saying that every finite étale cover of an abelian variety is again an abrlian variety, is that true? And do you have an explicit/geometri isomorphism between the two groups? $\endgroup$ – Filippo Alberto Edoardo Jun 15 '17 at 7:07
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    $\begingroup$ Yes, this is in Mumford, Abelian Varieties, Chapter IV. $\endgroup$ – user19475 Jun 15 '17 at 7:33
  • $\begingroup$ @FilippoAlbertoEdoardo Timo already gave a reference. If you look at the topological fundamental group of a complex torus, the result looks quite reasonable. $\endgroup$ – Felipe Voloch Jun 15 '17 at 8:30

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