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
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