Pointless curves of genus three and four
Number Theory
2010-01-23 v1 Algebraic Geometry
Abstract
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.
Cite
@article{arxiv.math/0403178,
title = {Pointless curves of genus three and four},
author = {Everett W. Howe and Kristin E. Lauter and Jaap Top},
journal= {arXiv preprint arXiv:math/0403178},
year = {2010}
}
Comments
LaTeX, 15 pages