Recovering affine curves over finite fields from $L$-functions

TL;DR

This paper explores how to reconstruct function fields over finite fields from their associated $L$-functions, providing explicit methods for recovering the structure of the fields and their defining equations.

Contribution

It introduces new techniques to recover affine curves over finite fields from $L$-functions of Galois extensions, including explicit recovery of points and equations.

Findings

Recovery of removed points on the projective line using $L$-functions of ray class fields.

Effective reconstruction of plane curve equations from Artin-Schreier extension $L$-functions.

Methodology applicable to various types of function fields over finite fields.

Abstract

Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$.