2008/09/07 by Theo van den Bogaart, Bogaart, Theo van den · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Computational Techniques in Science and Engineering #Advanced Data Processing Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.0809.1243
This is a (fully independent) chapter of the author's PhD thesis
arxiv created 2008/09/07 · openalex publication_date 2008/09/07 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic lift of the curve. The Frobenius morphism of the curve induces an automorphism of this cohomological space. The key step of the algorithm is to determine this automorphism with a sufficiently high p-adic precision: it is given in the form of a matrix with respect to a certain basis. Edixhoven has found a basis that has the property that the coefficients of the matrix are p-adically integral. This allows a smaller required precision, because a (semi-linear) power of this matrix must be computed up to some given precision. This text describes Edixhoven's basis and provides a proof of the fact that the basis is suitable.