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Computing Zeta Functions of Nondegenerate Curves

Wouter Castryck·Jan Denef·Frederik Vercauteren
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Abstract

In this paper 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 all known cases, e.g. hyperelliptic, superelliptic and C_{ab} 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 F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space complexity amounts to O(n^3g^4), assuming p is fixed.

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Wouter Castryck, Jan Denef, Frederik Vercauteren. 2007-01-08. Computing Zeta Functions of Nondegenerate Curves. https://arxiv.org/abs/math/0607308

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