Computing Zeta Functions of Nondegenerate Curves

2006/07/13 by Wouter Castryck, Castryck, Wouter, Jan Denef +3 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.math/0607308

openalex publication_date 2006/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_pⁿ, the expected running time is O(n³g⁶ + n²g^6.5), whereas the space complexity amounts to O(n³g⁴), assuming p is fixed.

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• Complexity of counting points on curves and the factor P₁(T) of the zeta function of surfaces

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