The Stacks project

64.32 Counting points

Let $X$ be a smooth, geometrically irreducible, projective curve over $k$ and $q = \# k$. The trace formula gives: there exists algebraic integers $w_1, \ldots , w_{2g}$ such that

\[ \# X(k_ n) = q^ n - \sum \nolimits _{i = 1}^{2g_ X} w_ i^ n + 1. \]

If $\sigma \in \text{Aut}(X)$ then for all $i$, there exists $j$ such that $\sigma (w_ i)=w_ j$.

Riemann-Hypothesis. For all $i$ we have $|\omega _ i| = \sqrt{q}$.

This was formulated by Emil Artin, in 1924, for hyperelliptic curves. Proved by Weil 1940. Weil gave two proofs

  • using intersection theory on $X \times X$, using the Hodge index theorem, and

  • using the Jacobian of $X$.

There is another proof whose initial idea is due to Stephanov, and which was given by Bombieri: it uses the function field $k(X)$ and its Frobenius operator (1969). The starting point is that given $f\in k(X)$ one observes that $f^ q - f$ is a rational function which vanishes in all the $\mathbf{F}_ q$-rational points of $X$, and that one can try to use this idea to give an upper bound for the number of points.

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