Pointless curves of genus three and four

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.

LaTeX, 15 pages

References in corpus (1)

• Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields

Cited by in corpus (2)

• Semiample Bertini theorems over finite fields
• An upper bound for the genus of a curve without points of small degree