On rationality of the intersection points of a line with a plane quartic

Details On rationality of the intersection points of a line with a plane quartic On rationality of the intersection points of a line with a plane quartic

2010-06-04

arxiv.org/abs/1006.0873v2

Mathematics

Number Theory

17 pages. Theorem 2 now includes the characteristic 2 case; Conjecture 1 from the previous version is proved wrong

Scientific paper

We study the rationality of the intersection points of certain lines and smooth plane quartics C defined over F_q. For q \geq 127, we prove the existence of a line such that the intersection points with C are all rational. Using another approach, we further prove the existence of a tangent line with the same property as soon as the characteristic of F_q is different from 2 and q \geq 66^2+1. Finally, we study the probability of the existence of a rational flex on C and exhibit a curious behavior when the characteristic of F_q is equal to 3.

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