Vol. 314, No. 1, 2021

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

Jeremy Booher and José Felipe Voloch

Vol. 314 (2021), No. 1, 1–28
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
Primary: 11G20
Milestones
Revised: 5 July 2021
Accepted: 11 July 2021
Published: 15 October 2021
Authors
 Jeremy Booher School of Mathematics and Statistics University of Canterbury Christchurch New Zealand José Felipe Voloch School of Mathematics and Statistics University of Canterbury Christchurch New Zealand