Pointless curves of genus three and four

Details Pointless curves of genus three and four Pointless curves of genus three and four

2004-03-10

arxiv.org/abs/math/0403178v1

pp. 125-141 in: Algebra, Geometry, and Coding Theory (AGCT 2003) (Y. Aubry and G. Lachaud, eds.), Seminaires et Congres 11, So

Mathematics

Number Theory

LaTeX, 15 pages

Scientific paper

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.

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