Drinfeld modular curves with many rational points over finite fields

2011
Cam, Vural
In our study Fq denotes the finite field with q elements. It is interesting to construct curves of given genus over Fq with many Fq -rational points. Drinfeld modular curves can be used to construct that kind of curves over Fq . In this study we will use reductions of the Drinfeld modular curves X_{0} (n) to obtain curves over finite fields with many rational points. The main idea is to divide the Drinfeld modular curves by an Atkin-Lehner involution which has many fixed points to obtain a quotient with a better #{rational points} /genus ratio. If we divide the Drinfeld modular curve X_{0} (n) by an involution W, then the number of rational points of the quotient curve W\X_{0} (n) is not less than half of the original number. On the other hand, if this involution has many fixed points, then by the Hurwitz-Genus formula the genus of the curve W\X_{0} (n) is much less than half of the g (X_{0}(n)).
Citation Formats
V. Cam, “Drinfeld modular curves with many rational points over finite fields,” Ph.D. - Doctoral Program, Middle East Technical University, 2011.