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kiseki
kiseki
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Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).

Does the abelianization of geometrically pro-$\ell$ etale fundamental group $\pi_{1}(A)^{\ell}$$(\pi_{1}(A\otimes\overline K)^{\ell})^{ab}$ isomorphic to the $\ell$-adic Tate module of $A$?

Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).

Does the pro-$\ell$ etale fundamental group $\pi_{1}(A)^{\ell}$ isomorphic to the $\ell$-adic Tate module of $A$?

Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).

Does the abelianization of geometrically pro-$\ell$ etale fundamental group $(\pi_{1}(A\otimes\overline K)^{\ell})^{ab}$ isomorphic to the $\ell$-adic Tate module of $A$?

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kiseki
kiseki
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pro-$\ell$ etale fundamental group of a semi-abelian variety

Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).

Does the pro-$\ell$ etale fundamental group $\pi_{1}(A)^{\ell}$ isomorphic to the $\ell$-adic Tate module of $A$?