Curves with many rational points via Atkin-Lehner involution

Official URL: https://dx.doi.org/10.1007/s40590-025-00746-4

In this article, we use reductions of the Drinfeld modular curves X0(n) to obtain curves over finite fields Fq of a given genus with many Fq-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 X0(n) by an involution W, then the number of rational points of the quotient curve W\X0(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\X0(n) is much less than half of the g(X0(n)).