Computing zeta functions of nondegenerate curves

Computing zeta functions of nondegenerate curves

We present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since in practice all known cases, for example, hyperelliptic, superelliptic, and Cab curves, can be transformed to f...

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Veröffentlicht in:International Mathematics Research Papers 2006-01, Vol.2006 (18), p.1-57
Hauptverfasser: Castryck, W., Denef, J., Vercauteren, F.
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Vercauteren, F.
description We present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since in practice all known cases, for example, hyperelliptic, superelliptic, and Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over Fpn, the expected running time is Õ(n3g6 + n2g6.5), whereas the space complexity amounts to Õ(n3g4), assuming p is fixed.
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title Computing zeta functions of nondegenerate curves
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