On Stöhr-Voloch's proof of Weil's theorem
Let \(X\) be a smooth, projective curve of genus \(g\) defined over the finite field \(\mathbb{F}_ q\). For any \(n\in \mathbb{N}\), let \(N_ n\) denote the cardinality of \(X(\mathbb{F}_{q^ n})\). Then \(X\) satisfies the Riemann Hypothesis (RH): \(| N_ n - q^ n - 1 | \leq 2g\cdot q^{n/2}\), whose proof is reduced to the following single inequality: \(N_ 1 \leq 1+q+2g\sqrt q\). Stöhr and Voloch have proved a stronger inequality: There exists a finite extension field \(k=\mathbb{F}_ q\) such that for sufficiently large \(n\), \(N_1 \leq (n-1)(g-1) + (q+n)(n+g) / n\). --- (RH) follows by taking \(q=n^2\) [see \textit{K.-O. Stöhr} and \textit{J. F. Voloch}, Proc. Lond. Math. Soc., III. Ser. 52, 1-19 (1986; Zbl 0593.14020).] In this paper, a simpler proof is given to the Stöhr-Voloch inequality. The idea of the proof is described as follows: There is a finite extension field \(k=\mathbb{F}_ q\) such that for \(n\) large enough, \(X\) embeds in \(\mathbb{P}^ n\) in such a way that the set of points \(x\) of \(X\) mapped via Frobenius into the osculating hyperplane of \(X\) at \(x\) is the support of a positive divisor, and that the number \(N\) of points in \(X(k)\) satisfies the Stöhr-Voloch inequality.
scientific article; zbMATH DE number 4127377
scientific article