Recovering affine curves over finite fields from \(L\)-functions. (English) Zbl 1522.11065
Summary: 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.
MSC:
| 11G20 | Curves over finite and local fields |
Keywords:
\(L\)-functions; function fields; curves; finite fields; recovering function fields; ray class field; Artin-Schreier coverReferences:
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