235,301 research outputs found
Sieving for rational points on hyperelliptic curves
We give a new and efficient method of sieving for rational points
on hyperelliptic curves. This method is often successful in proving that a
given hyperelliptic curve, suspected to have no rational points, does in fact
have no rational points; we have often found this to be the case even when our
curve has points over all localizations Qp. We illustrate the practicality of the
method with some examples of hyperelliptic curves of genus 1
Weierstrass semigroups on the Skabelund maximal curve
In 2017, D. Skabelund constructed a maximal curve over as
a cyclic cover of the Suzuki curve. In this paper we explicitly determine the
structure of the Weierstrass semigroup at any point of the Skabelund curve.
We show that its Weierstrass points are precisely the
-rational points. Also we show that among the Weierstrass
points, two types of Weierstrass semigroup occur: one for the
-rational points, one for the remaining
-rational points. For each of these two types its Ap\'ery set
is computed as well as a set of generators
On the maximum number of rational points on singular curves over finite fields
We give a construction of singular curves with many rational points over
finite fields. This construction enables us to prove some results on the
maximum number of rational points on an absolutely irreducible projective
algebraic curve defined over Fq of geometric genus g and arithmetic genus
Rational points on curves over function fields
We provide in this paper an upper bound for the number of rational points on
a curve defined over a one variable function field over a finite field. The
bound only depends on the curve and the field, but not on the Jacobian variety
of the curve
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