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Abstract
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Let
be an algebraic curve over a finite field of odd characteristic. We investigate using
-functions of Galois extensions
of the function field
of to effectively
recover the curve
.
When
is the projective line with four rational points removed, we show how to use
-functions of a
ray class field of
to effectively recover the removed points up to automorphisms of the projective line.
When
is a plane curve, we show how to effectively recover the equation of
using
-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
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Mathematical Subject Classification
Primary: 11G20
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Milestones
Received: 18 February 2021
Revised: 5 July 2021
Accepted: 11 July 2021
Published: 15 October 2021
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