Recovering affine curves over finite fields from L-functions

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.

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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

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

Vol. 314, No. 1, 2021

Jeremy Booher and José Felipe Voloch

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors