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International Mathematics Research Papers (2006) Vol. 2006 : article ID 72017, 57 pages, doi:10.1155/IMRP/2006/72017
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Computing zeta functions of nondegenerate curves

W. Castryck, J. Denef and F. Vercauteren

Department of Mathematics, University of Leuven Celestijnenlaan 200B, B-3001 Leuven-Heverlee, Belgium E-mail address: wouter.castryck{at}wis.kuleuven.be
Department of Mathematics, University of Leuven Celestijnenlaan 200B, B-3001 Leuven-Heverlee, Belgium E-mail address: jan.denef{at}wis.kuleuven.be
Department of Electrical Engineering, University of Leuven Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium E-mail address: frederik.vercauteren{at}esat.kuleuven.be

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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