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Weierstrass points and curves over finite fields. (English) Zbl 0593.14020
Let \(X\) be a projective, non-singular curve of genus \(g\) over a field with \(q\) elements and let \(N\) be the number of its rational points. In 1948, Weil established the Riemann hypothesis for such curves which implies that \(| N-(q+1)| \leq 2gq^{1/2}\). Recently, in a number of instances, it was shown that this bound can be improved. In the paper under review the authors develop a general approach to improving this bound based on an analysis of Weierstrass points. Examples are given and the relationship with the work of Stepanov and Weil is discussed.

14G15 Finite ground fields in algebraic geometry
11G20 Curves over finite and local fields
14G05 Rational points
14H55 Riemann surfaces; Weierstrass points; gap sequences
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