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 $v \nmid \ell$).
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2$\begingroup$ 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?). $\endgroup$– nfdc23Sep 27, 2017 at 13:08
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1$\begingroup$ Serre asserts this in "Examples" on page I-12 of Abelian $\ell$-adic representations and elliptic curves. $\endgroup$– Jeff YeltonSep 27, 2017 at 14:05
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$\begingroup$ @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. $\endgroup$– nfdc23Sep 27, 2017 at 18:18
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1 Answer
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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.
answered Oct 2, 2017 at 7:57