Rational Points on Rational Curves

TL;DR

This paper extends the relationship between $L$-functions and geometric properties from elliptic curves to rational curves, establishing new $L$-type series that encode geometric information through special values.

Contribution

It introduces two new $L$-type series for rational curves, linking their special values at 1 to geometric invariants, generalizing the elliptic curve $L$-function principle.

Findings

One $L$-type series matches the Dirichlet $L$-series of modulo 4, evaluating to $rac{ ext{pi}}{4}$ at 1.

The other $L$-type series's value at 1 equals a real period of the rational curve.

The series reflect the geometry of the rational curve through their special values.

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.