Theorem 63.27.4 (Deligne, Weil II). For a sheaf $\mathcal{F}_\rho$ with $\rho$ satisfying the conclusions of the conjecture above then the eigenvalues of $\pi _ X^*$ on $H_ c^ i(X_{\overline{k}}, \mathcal{F}_{\rho })$ are algebraic numbers $\alpha$ with absolute values

$|\alpha |=q^{w/2}, \text{ for }w\in \mathbf{Z},\ w\leq i$

Moreover, if $X$ smooth and proj. then $w = i$.

Proof. See [WeilII]. $\square$

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