# independence of $\ell$ of characteristic polynomial of Frobenius on $\ell$-adic Tate module of Abelian varieties over number fields

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 $v \nmid \ell$).

• With $v \nmid \ell$, see Application 1 in section 4 of math.stanford.edu/~conrad/BSDseminar/Notes/L3.pdf Hopefully someone knows a published reference (e.g., somewhere in Exp. IX of SGA7?). Sep 27 '17 at 13:08
• Serre asserts this in "Examples" on page I-12 of Abelian $\ell$-adic representations and elliptic curves. Sep 27 '17 at 14:05
• @JeffYelton: Those "Examples" in Serre's book don't address the characteristic polynomial on the space of inertial invariants at bad $v$ (as one varies $\ell$), which seems to be the main focus of the question. Sep 27 '17 at 18:18

## 1 Answer

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.