TITLE: Equivalent statements of the Riemann hypothesis in the Weil conjectures
QUESTION [18 upvotes]: In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of Frobenius acting on the $H^i_\mathrm{et}(X, Q_\ell)$ are algebraic integers with complex absolute value $q^{i/2}$.
For smooth proper curve $ C$ over $F_q$ of genus g, the Riemann hypothesis is often stated in another way as $|N- q - 1| \leq 2g \sqrt(q)$  where $N$ is the number of rational points on $ C$.
Why are these two statements equivalent?  Is (are) there any corresponding inequality(ies) which are equivalent to the Riemann hypothesis in higher dimensions?

REPLY [6 votes]: Regarding the question about higher dimensions, Scholl showed that RH in all dimensions follows from the statement:

If $X$ is a smooth hypersurface in $\mathbb{P}^{d+1}$ over $\mathbb{F}_q$, then $X(\mathbb{F}_{q^n}) - \mathbb{P}^d(\mathbb{F}_{q^n}) = O(q^{dn/2})$.

Here the $O(\ )$ is as $n \to \infty$, with the constant depending on $X$.