# Relation - Anabelian geometry and Tate conjecture

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"

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 :-).
• 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? – Filippo Alberto Edoardo Jun 15 '17 at 7:07