How do we deduce the following statement from the Chebotarev density theorem? The statement is from Ngo's Fundamental Lemma paper.



> 
> Let $U$ be a scheme of finite type over $\mathbb{F}\_q$. Let $\mathcal{L}\_1$, $\mathcal{L}\_2$ be two local systems of pure weight $i$. Suppose for any $u \in U$ and $k \in \mathbb{N}$, we have
> $Tr(\sigma\_u^k,\mathcal{L}\_1)=Tr(\sigma\_u^k,\mathcal{L}\_2)$, where $\sigma\_u$ is the Frobenius conjugacy class of $u$ in $\pi\_1(U)$.
> Then $\mathcal{L}\_1$ and $\mathcal{L}\_2$ are isomorphic up to semisimplification.
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> 
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Chebotarev density theorem and pure weight local systems
As Piotr says, we must assume $U$ normal. The purity assumption is not needed.


There are two steps to this proof


(1) Suppose for any $u \in U$ and $k \in \mathbb{N}$, we have
$Tr(\sigma\_u^k,\mathcal{L}\_1)=Tr(\sigma\_u^k,\mathcal{L}\_2)$, where $\sigma\_u$ is the Frobenius conjugacy class of $u$ in $\pi\_1(U)$. Then for any $\sigma \in \pi\_1(U)$, we have $Tr(\sigma^k,\mathcal{L}\_1)=Tr(\sigma^k,\mathcal{L}\_2)$


(2) Suppose that for any $\sigma \in \pi\_1(U)$, we have $Tr(\sigma^k,\mathcal{L}\_1)=Tr(\sigma^k,\mathcal{L}\_2)$ Then $\mathcal{L}\_1$ and $\mathcal{L}\_2$ are isomorphic up to semisimplification.


The first one follows from the Chebotarev density theorem. Because the traces are $\ell$-adic numbers, it suffices to prove the identity mod $\ell^n$ for each $n$. Whether this is true or not for a given $\sigma$ depends only on the image of $\sigma$ in some finite group (a subgroup of $GL\_m (\mathbb Z/\ell^n\mathbb Z)^2$ where $m$ is the rank of these representations).


But in a finite quotient of a Galois group (here is where we use normalcy, to get that the fundamental group is a quotient of a Galois group), every element arises from a Frobenius conjugacy class. Since we have the identity on every Frobenius element, we have it on every element.


The second one is a general fact about representations of groups over fields of characteristic zero. In particular, we do not use the $\ell$-adic topology, so we may assume the base field is $\mathbb C$. Let $G$ be the Zariski closure of the image of $\pi\_1$ in $GL\_m \times GL\_m$ acting on the two (semisimplified) representations. Because $G$ has a faithful semisimple representation, it is reductive. Take a maximal compact subgroup. If two representations are isomorphic on restriction to this compact subgroup they are isomorphic (Weyl). Now use character theory on this compact subgroup.


