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

Jeremy Booher and José Felipe Voloch

Vol. 314 (2021), No. 1, 1–28

DOI: 10.2140/pjm.2021.314.1

Abstract

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

Keywords

L-functions, function fields, curves, finite fields, recovering function fields, ray class field, Artin–Schreier cover

Mathematical Subject Classification

Primary: 11G20

Milestones

Received: 18 February 2021

Revised: 5 July 2021

Accepted: 11 July 2021

Published: 15 October 2021