WEIERSTRASS POINTS
AND
CURVES
OVER FINITE FIELDS
KARLOTTO STOHR
and
JOSE FELIPE VOLOCH
[Received 5 February 1985]
ABSTRAC T
For any projective embedding of a nonsingular 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 ordersequence 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, DrinfeldVladut 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 ordersequence 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), 119.
5388.3.52
A
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KARLOTTO 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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 Fall '10
 Clark
 Algebra, Riemann hypothesis, Algebraic curve, JOSE FELIPE VOLOCH

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