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WEIERSTRASS POINTS AND CURVES OVER FINITE FIELDS KARL-OTTO STOHR and JOSE FELIPE VOLOCH [Received 5 February 1985] ABSTRAC T For any projective embedding of a non-singular irreducible complete algebraic curve defined over a finite field, we obtain an upper bound for the number of its rational points. The constants in the bound are related to the Weierstrass order-sequence associated with the projective embedding. The bounds obtained lead to a proof of the Riemann hypothesis for curves over finite fields and yield several improvements on it. 0. Introduction Let X be a curve of genus g defined over a field k with q elements, and let N be the number of rational points of X. In 1948 Weil [12] proved the Riemann hypothesis for curves over finite fields which states that and, in particular, that N^q+\+2gq*. (*) Fixing g and making extensions of the constant field, we know that the above bound is the best possible in the sense that 2g cannot be replaced by a smaller constant. On the other hand, there are several instances in which (*) can be improved. The first result along these lines is due to Stark [10] in the hyperelliptic case, using Stepanov's method. Afterwards, Drinfeld-Vladut and Serre obtained improvements on (*) when g > ^(q — q*), using 'explicit formulae' (see [9] and the references therein). Serre also remarked that Weil's bound can be improved in general to where [•] denotes the integral part. The purpose of this paper is to give a general approach to the problem of improving (*). The idea is as follows. Consider X as embedded in some projective space. Using the equations for the osculating hyperplanes at the points on the curve, we define a function which vanishes at those points P whose images under the Frobenius map lie on the osculating hyperplane at P. This function will have zeros of high order at the rational points of X and a controlled number of poles. We thus get an upper bound for N which depends on q,g, the dimension of the ambient projective space, the degree of X, and on the Weierstrass order-sequence of the embedding. By The research of the second author is supported by CNPq (Brazil), Grant No. 200.916/82, and by ORS (England). A.M.S. (1980) subject classification: 14G15. Proc. London Math. Soc. (3), 52 (1986), 1-19. 5388.3.52 A
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2 KARL-OTTO STOHR AND JOSE FELIPE VOLOCH an appropriate choice of the embedding, we prove the Riemann hypothesis and, in several cases, we obtain improvements on (*). Our approach has some similarities with Stepanov's method (see [1, 7]). The fundamental difference is that, instead of obtaining an auxiliary function by solving linear equations, we give it explicitly in a conceptual way as a sort of wronskian determinant. Our approach has an interesting connection with Weil's original approach. This is
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