I am looking for a reference for the independence of $\ell$ of the characteristic polynomial of the Frobenius $\mathrm{det}(1-|\kappa(v)|^{-s}\mathrm{Frob}\_v \mid (V\_\ell A)^{I\_v})$ acting on the $\ell$-adic Tate module of an Abelian variety $A$ over a number field (you may assume $v \nmid \ell$).



independence of $\ell$ of characteristic polynomial of Frobenius on $\ell$-adic Tate module of Abelian varieties over number fields
This follows from Grothendieck's semistable reduction theorem -- the precise reference is SGA 7, Exp. IX, Thm 4.3(b).


The idea is to express the characteristic polynomial of Frobenius in terms of the special fiber of the Néron model of $A$ and then to write this special fiber as an extension of an abelian variety $B$ by an algebraic group $G$ which is itself an extension of a unipotent group $U$ by a torus $T$. We are thus reduced to show the independence of $\ell$ for $B$ and $T$, which basically follows from the Weil conjectures.


