5 research outputs found
Pointless Hyperelliptic Curves
In this paper we consider the question of whether there exists a hyperelliptic curve of genus g which is defined over but has no rational points over for various pairs . As an example of such a result, we show that if p is a prime such that is also prime then there will be pointless hyperelliptic curves over of every genus
On the linear bounds on genera of pointless hyperelliptic curves
An irreducible smooth projective curve over is called
\textit{pointless} if it has no -rational points. In this paper
we study the lower existence bound on the genus of such a curve over a fixed
finite field . Using some explicit constructions of
hyperelliptic curves, we establish two new bounds that depend linearly on the
number . In the case of odd characteristic this improves upon a result of R.
Becker and D. Glass. We also provide a similar new bound when is even
Pointless curves of genus three and four
A curve over a field k is pointless if it has no k-rational points. We show
that there exist pointless genus-3 hyperelliptic curves over a finite field F_q
if and only if q < 26, that there exist pointless smooth plane quartics over
F_q if and only if either q < 24 or q = 29 or q = 32, and that there exist
pointless genus-4 curves over F_q if and only if q < 50.Comment: LaTeX, 15 page