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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.

MSC:
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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