Rational Points on Rational Curves

Abstract

For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use all the counting information to define two $L$-type series. Then we consider special values of these series at $1$. One of the $L$-type series matches the Dirichlet $L$-series of modulo $4$, so the evaluation at $1$ is $\pi/4$; the special evaluation at $1$ of the other $L$-type series is equal to a real period associated to the rational curve. This identity confirms the general principle that an $L$-type series associated to a variety can reflect its geometry.